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<p id="mathjaxlink" class="pcenter"><a href="chap18.html">[MathJax off]</a></p>
<p><a id="X7DFC03C187DE4841" name="X7DFC03C187DE4841"></a></p>
<div class="ChapSects"><a href="chap18_mj.html#X7DFC03C187DE4841">18 <span class="Heading">Cyclotomic Numbers</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18_mj.html#X79E25C3085AA568F">18.1 <span class="Heading">Operations for Cyclotomics</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X8631458886314588">18.1-1 E</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X863D1E017BC9EB7F">18.1-2 Cyclotomics</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a 
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X869750DA81EA0E67">18.1-4 IsIntegralCyclotomic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X7DD6B95F79321D23">18.1-5 Int</a></span>
<span class="ContSS"><br /><span class="nocss">  </span>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X815D6EC57CBA9827">18.1-7 Conductor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X81DD58BB81FB3426">18.1-8 AbsoluteValue</a></span>
<class="ContSS"<br class=nocss"&;;span>a =chap18_mj.#X7808ECF37AA9004D"18.-9/>/span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X7AE2933985BE4C3E">18.1-10 CoeffsCyc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X803478CA7D2D830F">18.1-11 DenominatorCyc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X785F2CAB805DE1BE">18.1-12 ExtRepOfObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X7DDD51B983D5BC44">18.1-13 DescriptionOfRootOfUnity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 0
<span class"ContSS"> ><class=n"&;nbsp;/span>a href=chap18_mj.#E6CF4947D0A56F7".-5 sGaussRat/>/span
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X7FE3D5637B5485D0">18.1-16 java.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 0
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> <//span><a href="chap18_mj.html#X7EE5FB7181125E02">18.2 <span class="Heading">Infinity and negative Infinity</span></a>
<span>
<div class="ContSSBlock">
<span class="ContSS"><java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
</<iv class=example>pre
<div class="ContSect"><span class="<pan class="APprompt>&t/ <class"Int( (5+1/2E(5)^ ) ( 23E()3/2*(4 )/span>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class-4java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
</span>
<div ">
<span class="ContSS"><br /><span class="nocss">  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</span>
<span class="ContSS"><br /><span class="nocss">  <div class=func>tableclass=func ="100"<>tdclass=tdleft>codeclass"unc">&#8227;String(<=Arg></)/td><td class="tdright">( method )</td></tr></table></div>
</>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X8672D7F986CBA116">18.4-3 <span class="Heading">EY, EX, <span class="SimpleMath">\(\ldots\)</span>, ES</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X7E5985FC846C5201">18.4-4 <span class="Heading">EM, EL, <span class="SimpleMath">\(\ldots\)</span>, EJ</span>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</>
<span class="ContSS173"
<span <>/>
</div></div>
<div class<> ="815" name"815D6EC57CBA9827"/>/>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="<h5>18.1-7 Conductor</h5>
<span class="ContSS"><br /><span class="nocss">  </span><div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Conductor</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
spanclass=ContSS><br >< class="nocss">  </span><a href="chap18_mj.html#X7E361C057E97CA66">18.5-3 StarCyc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18_mj.html#X84438F867B0CC299">
<span class="ContSS"><br /><span class="nocss">  < =GAPpromptgap&;/> < class=GAPinput>Conductor(0 );Conductor (10 ;Conductor (2))<
<span class="ContSS"><java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 1
</div></div>
<div class="ContSect"><span class="tocline"><span <p><a id="X81DD58BB81FB3426" name="X81DD58BB81FB3426
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 0
<
</div>

<h3>18 <span class="Heading">Cyclotomic Numbers</span></h3>

< =pkg><> injava.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 545

<p>Information concerning operations for domains of cyclotomics, for example certain integral bases of fields of cyclotomics, can be found in Chapter <a href="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X79E25C3085AA568F" name="X79E25C3085AA568F"></a></p>

<h4>18.pre>/div

<p><a id="X8631458886314588" name="X8631458886314588"></a></pp><a id="X7808ECF37AA9004D" name="X7808ECF37AA9004D"></a></p>

<h5>18.1-1 E</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; E</code>( <var class="Arg">n</var> )</td><td class81EA0E67EA0E67"<span class="RefLink">18.1-4</span></a>)) near to the cyclotomic <var class="Arg">cyc</var> in the following sense: Let <code class="code">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><code class="func">E</code> returns the primitive <var class="Arg">n</var>-th root of unity <span class="SimpleMath">\(e_n = \exp(2\pi i/n)\)</span>. Cyclotomics are usually entered as sums of roots of unity, with rational coefficients, and irrational cyclotomics are displayed in such a way. (For special cyclotomics, see <a href="chap18_mj.html#X7B242083873DD74F"><span class="RefLink">18.4</span></a>.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">E(9); E(9)^3; E(6); E(12) / 3;</span>
-E(9)^4-E(9)^7
E(3)
-E(^2
-1/3*E(12)^7
</pre></div>

<p>A particular basis is used to express cyclotomics, see <a href=E()+E(5))^2

<p><a id="X863D1E017BC9EB7F" name="X863D1E017BC9EB7F"></a></p>

<h5>18.1-2 Cyclotomics</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Cyclotomics</code></td><td class="tdright">( global variable )</java.lang.StringIndexOutOfBoundsException: Index 177 out of bounds for length 12
<>    cyclotomics.</p>


<div class="example"><pre>
<span class=G>gapgt;/>< =GAPinput">(9) in ;37 inCyclotomics; true in Cyclotomics;</span>
true
true
false
</pre></div>

<p>As the cyclotomics are field elements, the usual arithmetic operators <code class="code">+</code>, <code class="code">-</code>, <code class="code">*</code> and <code class="code">/</code> (and <code class="code">^</code> to take powers by integers) are applicable. Note that <code class="code">^</code> does <em>not</em> denote the conjugation of group elements, so it is <em>not</em> possible to explicitly construct groups of cyclotomics. (However, it is possible to compute the inverse and the multiplicative order of a nonzero cyclotomic.) Also, taking the <span class="SimpleMath">\(k\)</span>-th power of a root of unity <pLet<var class=Arg>cycvar a with < class"impleMath"\n\<span> ( <codejava.lang.StringIndexOutOfBoundsException: Range [120, 119) out of bounds for length 1247


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">E(5) + E(3); (E(5) + E(5)^4) ^ 2; E(5) / E(3); E(5) * E(3);</span>
-E(15)^2-2*E(15)^8-E(15)^11-E(15)^13-E(15)^14
-2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
E(15)^13
E(15)^8
<span class="GAPprompt">gap></span> <span class="GAPinput">Order( E(5) ); Order( 1+E(5) );</span>
5
infinity
</pre></div>

<p><a id="X841C425281A6F775" name="X841C425281A6F775"></a></p>

<h5>18.1-3 IsCyclotomic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsCyclotomic</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsCyc</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Every object in the family <code class="code">CyclotomicsFamily</code> lies in the category <code class="func">IsCyclotomic</code>. This covers integers, rationals, proper cyclotomics, the object <code class="func">infinity</code> (<a href="chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>), and unknowns (see Chapter <a href="chap74_mj.html#X7C1FAB6280A02CCB"><span class="RefLink">74</span></a>). All these objects except <code class="func">infinity</code> (<a href="chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) and unknowns lie also in the category <code class="func">IsCyc</code>, <code class="func">infinity</code> (<a href="chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) lies in (and can be detected from) the category <code class="func">IsInfinity</code> (<a href="chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>), and unknowns lie in <code class="func">IsUnknown</code> (<a href="chap74_mj.html#X828556067E069B6D"><span class="RefLink">74.1-3</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomic(0); IsCyclotomic(1/2*E(3)); IsCyclotomic( infinity );</span>
true
true
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyc(0); IsCyc(1/2*E(3)); IsCyc( infinity );</span>
true
true
false
</pre></div>

<p><a id="X869750DA81EA0E67" name="X869750DA81EA0E67"></a></p>

<h5>18.1-4 IsIntegralCyclotomic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsIntegralCyclotomic</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A cyclotomic is called <em>integral</em> or a <em>cyclotomic integer</em> if all coefficients of its minimal polynomial over the rationals are integers. Since the underlying basis of the external representation of cyclotomics is an integral basis (see <a href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>), the subring of cyclotomic integers in a cyclotomic field is formed by those cyclotomics for which the external representation is a list of integers. For example, square roots of integers are cyclotomic integers (see <a href="chap18_mj.html#X7B242083873DD74F"><span class="RefLink">18.4</span></a>), any


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r:= ER( 5 );               # The square root of 5 ...</span>
E(5)-E(5)^2-E(5)^3+E(5)^4
<span class="GAPprompt"gap></span> <span class="APinput">IsIntegralCyclotomic( r ); # ... is a cyclotomic integer.</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">r2:= 1/2 * r;              # This is not a cyclotomic java.lang.StringIndexOutOfBoundsException: Index 123 out of bounds for length 111
1/2*E(5)-1/2*E[0 1,0 ,0 java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
<span java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 91
false
<span class="GAPprompt">gap></span> <span class="GAPinput">r3:= 1/2 * r[0,0 0 ,0 , ,0 -1 0,,0 0 ,-,0]
E(5)+E(5)^4
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIntegralCyclotomic( r3 );</span>
true
</pre></div>

<p><a id="X7DD6B95F79321D23" name="X7DD6B95F79321D23"></a></p>

<h5>18.1-5 Int</h5>

<class">table class="unc width="100%>tr><td class=tdleft"><code class="func">&#8227; Int</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p> operation <code class="func">Int</code> can be used to find a cyclotomic integer near to an arbitrary cyclotomic, by applying <code class="func">Int</code> (<a href="chap14_mj.html#X87CA734380B5F68C"><span class="RefLink">14.2-3</span></a>) to the coefficients.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Int( E(5)+1
E(5)
-E(4)
</pre></div>

<p><a id="span class="GAPprompt">gap></span> <span class="GAPinput">E(9);  DescriptionOfRootOfUnity( E(9) );</span>

<h5>18.- </h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class=
<p>a ="8712419182ECD8DD" name"8712419182ECD8DD"></a></p>


<div class="<5>8114IsGaussInt</5>
<span class=java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
"E(5)+1/2*E(5)^2"
"17/3"
</pre></div>

p>aid"8156EC57CBA9827" ="">/></>

<h5>18.1-7 Conductor</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleftjava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 0
<div class="div class=f">table class= width=100"<java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 217
<p>For an element <var class="Arg">cyc</var> of cyclotomicfield < class="func">Conductor</code> returns the smallest integer <span class="SimpleMath">\(n\)</span> such that <var class="Arg">cyc</var> is contained in the <span class="SimpleMath">\(n\)</span>-th cyclotomic field. For a collection <var class="Arg">C</var> of cyclotomics (for example a dense list of cyclotomics or a field of cyclotomics), <><code class="func">IsGaussRat</code> returns <code class="keyw">true</code> if the object <var class="Arg">x</var> is a Gaussian rational (see <code class="func">GaussianRationals</code> (<a href="chap60_mj.html#X82F53C65802FF551"><span class="RefLink">60.1-3</span></a>)), and <code class="keyw">false</code> otherwise. Gaussian rationals are of the form <span class="SimpleMath">\(a + b\)</span><code class="code


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Conductor( 0 ); Conductor( E(10) ); Conductor( E(12) );</span>
1
5
12
</pre></div>

<p><a id="X81DD58BB81FB3426" name="X81DD58BB81FB3426"></a></p>

<h5>18.1-8 AbsoluteValue</h5>

<div class="func"><table class= class="GAPprompt"gapgt< spanclass""Field ()+()4) ( E5+()4 ;/span>
<p>returns the absolute value of a(5, 1, 4]


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span classp>a id="" name"E"<a<p
3
</pre></div>

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0

<h5>18.1-9 RoundCyc</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class
<>is a cyclotomic <span="impleMath">(\)</span> ( < =func"</code>(<a =".html#DA81EA0E67< class""18.1-4</span>/>)nearto  varclass"cyc</ar inthe following sense:  <<class="code">c</code> be the <span class="SimpleMath">\(i\)</span>-th coefficient in the external representation (see <code class="func">CoeffsCyc</code> (<a href="chap18_mj.html#X7AE2933985BE4C3E"><span class="RefLink">18.1-10</span></a>)) of <var class="Arg">cyc</var>. Then the <span class="SimpleMath">\(i\)</span>-th coefficient in the external representation of <span class="SimpleMath">\(z\)</span> is <code class="code">Int( c + 1/2 )</code> or <code class="code">Int( c - 1/2 )</code>, depending on whether <code class="code">c</code> is nonnegative or negative, respectively.</p>

<p>Expressed in terms of the Zumbroich basis (see <a href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>), rounding the coefficients of <var class="Arg">cyc</var> w.r.t. this basis to the nearest integer yields the coefficients of <span class="SimpleMath">\(z\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RoundCyc( E(5)+1/2*E(5)^2 ); RoundCyc( 2/3*E(7)+3/2*E(4) );</span>
E(5)+E(5)^2
-2*E(28)^3+E(28)^4-2*E(28)^11-2*E(28)^15-2*E(28)^19-2*E(28)^23
 -2*E(28)^27
</pre></div>

<p><a id="X7AE2933985BE4C3E" name="X7AE2933985BE4C3E"></a></p>

<h5>18.1-10 CoeffsCyc</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CoeffsCyc</code>( <var class="Arg">cyc</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">cyc</var> be a cyclotomic


<div class="example"><pre>
< ""gapg<span<class"APinput>: (5)E5^;/span>
E(5)+E(5)^2
<span class="GAPprompt">gap></span> <span class="GAPinput">CoeffsCycp>Often it is useful to distinguish <code class="func">infinity</code> from <q>proper</q> cyclotomics. For that, <code class="func">infinity</code> lies in java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 0, 1, 1, 0, 0 ]
[ 0 -,0  ,, 
fail
</pre></div>

<p><a id="X803478CA7D2D830F" name="X803478CA7D2D830F"></a></p>

<h5>18.1-11 DenominatorCyc</h5>

<div class="func"><table class="func" width="100%"><tr><td
<pFor a cyclotomic number <var class="Arg">cyc</var> (see <code class="func">IsCyclotomic</code> (<a href="chap18_mj.html#X841C425281A6F775"><span class="RefLink">18.1-3</span></a>)), this function returns the smallest positive integer <span class="SimpleMath">\(n\)</span> such that <span class="SimpleMath">\(n\)</span><code class="code"> * </code><var class="Arg">cyc</var> is a cyclotomic integer (see <code class="func">java.lang.StringIndexOutOfBoundsException: Index 441 out of bounds for length 5

<p><a id="X785F2CAB805DE1BE" name="X785F2CAB805DE1BE"></a></p>

<h5>18.1-12 ExtRepOfObj</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ExtRepOfObj</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( method )</td></tr></table></div>
</var> with conductor <span =SimpleMath">n\/span (<class=func"Conductor<code> (<a href="chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>) is the list returned by <code class="func">CoeffsCyc</code> (<a href="chap18_mj.html#X7AE2933985BE4C3E"><span class="RefLink">18.1-10</span></a>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<div class="example">java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 137
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtRepOfObj( E(5) ); CoeffsCyc( E(5), 5 );</span>
[ 0, 1, 0, 0, 0 ]
[ 0, 1, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span true
[ 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0 ]
</pre></div>

<p><a id="X7DDD51B983D5BC44" name="X7DDD51B983D5BC44"></ah4>8.  class"eading"ATLASIrrationalities/>/h4

<h5>18.1-13 DescriptionOfRootOfUnity</h5>

<div class="h5>4-1 < ="HeadingEB EC,<="impleMath"\l\)span> <</>
<p>Given a cyclotomic <var class="Arg">root</var> that is java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">E(9);  DescriptionOfRootOfUnity( E(9) );</span>
-E(9)^4-E(9)^7
[9 1]
<span class="GAPprompt">gap></span> <span class="GAPinput">DescriptionOfRootOfUnity( -E(3) );</span>
[6  ]
</pre></div>

<p><a id="X8712419182ECD8DD" name="X8712419182ECD8DD"></a></p>

<h5>18.1-14 IsGaussInt</h5>

""table="width="100%"><tr><td class="tdleft"><code class="func">‣ IsGaussInt</code>( <var class="Arg">x</var> )</td><td class="tdright">( function )</td></tr></table></div>
<="html#80BD5EAB879F096E>span  class="efLink>0.-1</span</a))  <code=keyw>false</>otherwise.Gaussian integers are  theform <span ="java.lang.StringIndexOutOfBoundsException: Range [380, 379) out of bounds for length 533

<p><a id="X7E6CF4947D0A56F7" name="X7E6CF4947D0A56F7"></a></p>

<h5>18.1-15 IsGaussRat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGaussRat</code>( <var class="Arg">x</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">IsGaussRat</code> returns <code class="keyw">true</code> if the object <var class="Arg">x</var> is a Gaussian rational (see <code class="func">GaussianRationals</code> (<a href="chap60_mj.html#X82F53C65802FF551"><span class<div class=func>table =func "100"<>< class=tdleft"<code class="unc"ãEG</ode>( <arclass=Arg"Nvar>)td><tdclass=tdright"(n;function )/>/tr>/table></div>

<p><a id="X7FE3D5637B5485D0" name="X7FE3D5637B5485D0"></a></p>

<h5>18.1-16 DefaultField</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DefaultField</code>( <var class="Arg">list</
<p><code class="func">DefaultField</code> for cyclotomics is defined to return the smallest <em>cyclotomic</em> field containing the given elements.</p>

<p>Note that <code class="func">Field</code> (<=java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 120


class>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Field( E(5)+E(5)^4 );  DefaultField( E(5)+E(5)^4 );</span>
NF(5,[ 1, 4 ])
CF(5)
</pre></div>

<p><a id="X7EE5FB7181125E02" name="X7EE5FB7181125E02"></a></p>

< <span ="Heading">Infinity java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 73

<p><a id="X8511B8DF83324C27" name="X8511B8DF83324C27"></a></p>

h5. <

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 5
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsNegInfinity</code>( <var class="Arg">obj</var> )</td><td class="<d >>
<classtdleft>span =SimpleMath"\\eft(\um_{j  1}}{t{N}} ^{^}} \ight)/4\<span>td>
<div class=td"java.lang.StringIndexOutOfBoundsException: Range [21, 19) out of bounds for length 27
<p><java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 5

<p>Some arithmetic operations are provided <d =tdleft"><code class="code">EE(</code><var class="Arg">N</var><code class="code">)</code></td>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">-infinity + 1;</span>
-infinity
<span class="GAPprompt">gap></span> <span class="GAPinput">infinity + infinity;</span>
infinity
</pre></div>

<p>Often it is useful to distinguish <code class="func">infinity</code> from <q>proper</


<java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
<span class="GAPprompt">gap>class="dleft><pan class="SimpleMath">\(f_{\textit{N}}\)</span></td>
infinity
<span class="GAPprompt">gap<td class"tdcenter">=>
true
true
false
true
<span class=
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( [ s, 2, s, E(17), s, 19 ] );</span>
 ,,E17, ]
</pre></div>

<p><a id="X7F66A62384329705" name="X7F66A62384329705"></a></p>

<h4>18.3 <span class="Heading">Comparisons of Cyclotomics</span></h4>

<  cyclotomics,theoperators < class="odelt;</ode>, <code class="code"><=</code>, <code class="code">=</code>, <code class="code">>=</code>, <code class="code">></code>, and <code class="code"><></code> can be used, the result will be <code class="keyw">true</code> if the first operand is smaller, smaller or equal, equal, larger or equal, larger, or unequal, respectively, and <code class=<td class="tdcenter"></djava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

<p>Cyclotomics are ordered as follows: The relation between rationals is the natural one, rationals are smaller than irrational cyclotomics, and <code class="func"td=""=</

<p>For/>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span classjava.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 88
false
span=">><span><span>(3 <E()2    # both   3 compare the ext.repr./span>
false
<<div
true
true
</re</>

<p><a id="X7B242083873DD74F" name="X7B242083873DD74F"></a></p>

<h4>18.4 <span class="Heading">ATLAS Irrationalities</spanspan">/java.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 83

<p><a id="X8414ED887AF36359" name="X8414ED887AF36359"></a></p>pre>/>

<h5>18.4-1 <span class="Heading">EB, EC, <span class="SimpleMath">\(\ldots\)</span>, EH</span></h5>

< =f"="width"100"<><td=t"<code =f"&8227;/codevar""<var <<>&bsp;functionjava.lang.StringIndexOutOfBoundsException: Range [184, 183) out of bounds for length 209
<div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 0
< class"unc><able ="unc ="100"<>td class=t"< =func"&8227 </code> <ar=A"N</var )/>tdclass"tdright"(nbsp;nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EE</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EF</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
classf">java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 209
<="><table ""=100"<><td =tdleft<codeclass"unc"&#8227; <code( <var class""/var )/>td class="dright>(&;functionnbsp;</><tr>/table><div>
<p>For a positive integer <var class="Arg">N</var>, let <span class="SimpleMath">\(z =\)</span> <code class="code">E(</code><var class="Arg">N</var><code class="code">)</code> <span class="SimpleMath">\(= \exp(2 \pi i/\textit{N})\)</span>. The following so-called <em>atomic irrationalities</em> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, Chapter 7, Section 10]</a>) can be entered using functions. (Note that the values are not necessary irrational.)</p>

<div class="pcenter"><table class="GAPDocTablenoborder">
<tr>
<td class="tdleft"><code class="code">EB(</code><var java.lang.StringIndexOutOfBoundsException: Range [0, 58) out of bounds for length 0
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(b_{\textit{N}}\)</span></td>
<tddiv class"pcenter">tableclass=java.lang.StringIndexOutOfBoundsException: Range [56, 54) out of bounds for length 56
<td class="tdleft"><span class="SimpleMath">\(\left( \sum_{{j = 1}}^{{\textit{N}-1}} z^{{j^2tdjava.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 28
<td class="tdcenter">,</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{2}\)</span></td>
</
<tr>
<td class="tdleft"><code class="code">EC(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(c_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(\java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 0
<td class="tdcenter">,</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{3}\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">ED(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(d_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<</pre<div
<td class="tdcenter">,</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{4}\)</span></td<>< id"X8672D7F986CBA116"name"8672D7F986CBA116"<p>
/>
<tr>
< java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 104
<td class="tdcenter">=</td>
<divfunctable=""00>< java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 237
<td class=""=/td>
<td class="tdleft"><span class="SimpleMath">\(\left( \sum_{{j = 1}}^{{\textit{N}-1}} z^{{j^5}} \right) / 5\)</span></td>
<class"tdcenter",/td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{5}\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">EF(</code><var class="Arg">N</var><code class="code">)</code></td>
<dclass="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(f_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(\left( r><td class="tdleft"><code class="func">&#pan style='color: green'>8227; EU</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<td class="tdcenter">,</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{6}\)</span></td>
</tr>
<>
<td class="tdleft"><code class="code">EG(</code><var class="Arg">N</var><code class="code">)</code></td>
td class="dcenter><td>
<td class="tdleft"><span class="SimpleMath">\(g_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleftjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<td class="tdcenter">,</td>
<dclass="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{7}\)</span></td>
</tr>
<tr>
<td >definespan=S>( =\exp( \ t{N))/>/
<td class="tdcenter">=</td>
<d class=""< class=">\h_{N}\<><td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class \sum_{{j  1}^\textit{}-} ^{j^} \ight) / 8\)</span></td>
<td class="tdcenter">,</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{8}\)</span></td>
</tr>
</table><br />
</div>

<p>(Note that in <code class="code">EC(</code><var class="Arg">N</var><code class="code">)</code>, <span class="SimpleMath">\(\ldots\)</span>, <code class="code">EH(</code><var class="Arg">N</var><java.lang.StringIndexOutOfBoundsException: Range [0, 201) out of bounds for length 5


classexample>>
<span class="GAPprompt">gap></span> <span class="GAPinput">EB(5);  EB(9);</span>
E(5)+E(5)^4
1
</pre></div>

<p><a id="X813CF4327C4B4D29" name="X813CF4327C4B4D29"></a></p>=<java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 74

2="EI<>h5

<iv =f>< class=func =100"tr<td t" ">#; EI<> var=A"Nvar/d< =">nfunction)/>/tr>/<>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ER</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div
<> rational <var class=Arg>N/var><f"ER<>returns the square  <span =""\({\textitN}})/>of < class="Arg">N</var>, and <code class="func">EI</code> returns <span class="SimpleMath">\(\sqrt{{-\textit{N}}}\)</span>. By the chosen embedding of cyclotomic fields into the complex numbers, <code class="func">ER</code> returns the positive square root if <var class="Arg">N</var> is positive, and if <var class="Arg">N</var> is negative then <code class="code">ER(</code><var class="Arg"><d class="dcenter><td>

<p><code class="func">ER</code> is installed as method for the java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 27

<p>From a theorem of Gauss (see <a href="chapBib_mj.html#biBLan70">[Lan70, QS4 in Chtdtjava.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 100

<div class="pcenter"><table class="GAPDocTablenoborder">
<tr>
<td class="tdleft"><span "java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 104
<td class="tdcenter">if</td>
td class"dleft">span class=SimpleMath"\\{} \quiv  \pmod 4\)</panjava.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
</ =tdleft> =SimpleMath\t_\{}\)<span</td>
<tr>
<"><span =SimpleMath"\(1+i \sqrt{{{}}  2\)/span>/td>
<td class="tdcenter">if</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv -1 \pmod 4\)</span></td>
<tr>
</table><br />
</div>

<p>So <span class="SimpleMath">\(\sqrt{{\textit{N}}}\)</span> can be


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ER(3); EI(3);</span>
-E(12)^7+E(12)^11
E(3)-E(3)^2
</pre></div>

<p><a id="X8672D7F986CBA116" name="X8672D7F986CBA116"></a></p>

<h5>18.4-3 <span class="Heading">EY, EX, <span class="SimpleMath">\(\ldots\)</span>, ES

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EY</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function&java.lang.StringIndexOutOfBoundsException: Index 211 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EX</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EW</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EV</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EU</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ET</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ES</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>For the given integer <var class="Arg">N</var> <span class="SimpleMath">\(> 2\)</span>, let <span class="SimpleMath">\(\textit{N}_k\)</span> denote the first integer with multiplicative order exactly <span class="SimpleMath">\(k\)</span> modulo <var class="Arg">N</var>, chosen in the order of preference</p>

<p class="center">\[
1, -1, 2, -2, 3, -3, 4, -4, \ldots .
\]</p>

<p>We define (with <span class="SimpleMath">\(z = \exp(2 \pi i/\textit{N})\)</span>)</p>

<div class="pcenter"><table class="GAPDocTablenoborder">
<tr>
<td class="tdleft"><code class="code">EY(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(y_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(z + z^n\)</span></td>
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_2)\)</span></td>
</tr>
<tr>
td class="tdleft"><code class="code">EX(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<td class=tdleft>span "impleMath\x_{textit{}\</pan></>
<td class="tdcenter">=<>Let  class="rg">N</var> be an integer, <var class="Arg">N</var> <span class="SimpleMath">\(> 2\)</span>. We define (with <span class="SimpleMath">\(z = \exp(2 \pi i/\textit{N})\)</span>)</p>
<td class="tdleft"><span class="SimpleMath">\(z + z^n + z^{{n^2}}\)</span></td>
<d class=tdleft"<span class="impleMath>(n=\textit{N}_3)\)</span></td>
</tr>
<tr>
<td java.lang.StringIndexOutOfBoundsException: Range [78, 9) out of bounds for length 78
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(w_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<d class="tdleft"><span class="SimpleMath">\(z + z^n + z^{{n^2}} + z^{{n^3}}\)</span></td>
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_4)\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">EV(</code><var class="java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 27
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(v_{\textit{N}}\)d"java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 78
<td classb/
<td class="tdleft"><span class="SimpleMath">\(z + z^n + z^{{n^2}} + z^{{n^3}} + z^{{n^4}}\)</span></td>
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_5)\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">EU(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<<p>"844" =><>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(z + <td class="tdleft"><span class="SimpleMath">\(z + z
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_6)\)</span></td>
/tr>
<tr>
<td class="tdleft"><code class="code">ET(</code><var class=p>Let <span class="impleMath">\(\textit{N}_{\textit{k}}^{(\textit{d})}\)</span> be the <span class="SimpleMath">\((\textit{d}+1)\)</span>-th integer with multiplicative order exactly <var class="Arg">k</var> modulo <var class="Arg">N</var>, chosen in the order of preference defined in Section <a href="chap18_mj.html#X8672D7F986CBA116"><span class="RefLink">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(t_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(z + z^n + z^{{n^2}} + \ldots + z^{{n^6java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<td< class="enter"\[
</tr>
<tr>
<td class="tdleft"><code class="code">ES(y_< =Arg"N<var}{prime\ ={var class=Arg"N/var>{2} \dots,
<td class="tdcenter">=</td>
eft>spanclass=SimpleMath>(\{}\<span>/java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(z + z^n + z^{{n^2}} + \ldots + z^{{n^7}}\)</span></td>
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_8)\)</span></td>
<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
</table><br />
</iv>

<>Forthetwo-argument versions of the functions, see Section <code class="func">NK</code> (<a href="chap18_mj.html#X844F0EBF849EDEB3"><span class="RefLink">18.4-5</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="class">span classS">y{tN}^(textit{)\/>java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
E(5)+E(5)^4
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 88
0
E(17)+Etd "tdcenter"=/td>
</pre></div>

<p><a id="X7E5985FC846C5201" name="X7E5985FC846C5201"></a><r>

<h5>18.4-4 <span class="Heading">EM, EL, <span class="SimpleMath">\(\ldots\)</span>, EJ</span<td ">/>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EM</code>( <var class="Arg">N</var>[, <var classtd ="tdleft"></td>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class=</tr>
<iv class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#n style='color: green'>8227; EK</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EJ</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<pleMath">\(> 2\)</span>. We define (ith s classSimpleMath>( =\(2 \pi i/\textit{N}\)</pan>)p>

<div class="pcenter"><table class="GAPDocTablenoborder">
<tr>
<td class="tdleft"><code class="code">EM(</code><var class="Arg">N</var><code class=/java.lang.StringIndexOutOfBoundsException: Range [6, 7) out of bounds for length 6
 =>t>
<td class="tdleft"><span class="SimpleMath">\(m_{\textit{N}}\)<>.- h>
<td class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
<td class="tdleft"><span class="SimpleMath">\(z - z^n\)</span></td>
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_2)\)</span></td>>et java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 948
</r>
<tr>
<td class="tdleft"><code class="code">EL(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(l_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(z - z^<div class="pcenter"><table lass="GAPDocTablenoborder">
<td class="tdleft"span =SimpleMath">( = textit{}_4)\)/span>/tdjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
</tr>
<tr>
td ="><>
<td class="tdcentertd ="dleft"<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 108
<td class="tdleft"><span class="SimpleMath">\(k_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(z - z^n +tdtdleft<ode class="code">4qN&3&5&7-3&4</code></td>
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_6)\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">EJ(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcentertd class=tdleft"><span class="SimpleMath">\(4 (q_N + q_N^{{*3}} + q_N^{{*5}} + q_N^{{*7}}) - 3 q_N^{{*11}}\)</span></td>
<td class="tdleft"><span class="SimpleMath">\(j_{\textit{N}}\)</span></td>
< ="tdcenter"=/td>
<td ="java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 88
<td class="tdleft"><span class="SimpleMath">\((tr>
</tr>
<table><br >
</div>

<p>For the two-argument versions of the functions, see Section <code<> explain the <q>ampersand</q> syntax in general we remarkexplain the <q>ampersand</q> syntax in general we remark that <q>&k</q> is interpreted as <span class="SimpleMath">\(q_N^{{*k}}\)</span>, where <span class="SimpleMath">\(q_N\)</span> is the most recently named atomic irrationality, and that the scope of any premultiplying coefficient is broken by a <span class="SimpleMath">\(+\)</spanjava.lang.StringIndexOutOfBoundsException: Index 347 out of bounds for length 0

<p><a id="X844F0EBF849EDEB3" name="X844F0EBF849EDEB3"></a></p>

<h5>18.4-5 NK</h5>

<div class="func"><table class="func" width="pan =""gap></span> <span class="GAPinput">AtlasIrrationality( "-3y'''24*13&5" );</span>
<p>Let <span class="SimpleMath">\(\textit{N}_{\textit{k}}^{(\textit{d})}*(8)3*(^3

<p>We write <span class="SimpleMath">\(\textit{N}_{\textit{k}} = \textit{N}_{\textit{k}}^{(0*(24)2*24^112E()17+*)java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

<p>The algebraic numbers</p>

<
y_{<var class="Arg">N</var>}^{\prime} = y_{<var class="Arg">N</var>}^{(1)},
y_{<var class="Arg">N</var>}^{\prime\prime} = y_{<var class="Arg">N</var>}^{(2)}, \ldots,
x_{<var classGclass"APprompt"gap&;/pan sjava.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 107
j_{<var class="Arg">N</var>}^{\prime}, j_{<var class="Arg">N</varE()6*24)^
\]</p>

<p>are obtained on replacing <span class="SimpleMath">\(\textit{N}_{\textit{k}}\)</span> in the definitions in the sections <a href="chap18_mj>a id="X79FE34337DF2CD10name=X79FE34337DF2CD10"><a><pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<div class="pcenter"><table class="GAPDocTablenoborder">
<tr>
<td class="tdleft"><code class="code">EY(</code><var classp>a id"name"X79EE9097783128C4>/<p>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(y_{\textit{N}}<518.51 GaloisCyc<h5>
</tr>
<tr>
<td class="tdleft"><code class="code">EX(</code><var class="Arg">N</var>,<var class="Arg">d</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(x_{\textit{N}}^{(\textit{d})}\)</span></td>
</tr>
<tr>
<td class="tdleft"></td>
<tdclass"tdcenter"><span class="SimpleMath">\(\ldots\)</span></td>
<td class="tdleft"></td>
</tr>
<tr>
<java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 129
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(j_{\textit{N}}^{(\textit{d})}\)</span></td>
</tr>
</<p><a id"X7BE001A0811CD599" name=X7BE001A0811CD599">/a>p>
</div>

<p><a id="X812E334E7A869D33" name="java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 32

<h5>18.4-6 AtlasIrrationality</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AtlasIrrationality</code>( <var class="Arg">irratname</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">irratname</var> be a string that describes an irrational value as a linear combination in terms of the atomic irrationalities introduced in the sections <a href="chap18_mj.html#X8414ED887AF36359"><span class="RefLink">18.4-1</span></a>, <a href="chap18_mj.html#X813CF4327C4B4D29"><span class=<div ""<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 221

ivclass"> class=GAPDocTablenoborder">
<tr>
<td class="tdleft"><code When <code class="func">ComplexConjugate called with a list then the result is the list of return values of <code class="func">ComplexConjugate</code> for the list entries in the corresponding positions.</p>
<td class="tdcenter">means</td>
<td class="tdleft"><span class="SimpleMath">\(2 q_N + 3 q_N^{{*5}} - 4 q_N^{{*7}} + q_N^{{*9}}\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">4qN&3&5&7-3&4</code></td>
<td class="tdcenter">means</td>
<td class="tdleft"><span class="SimpleMath">\(4 (q_N + q_N^{{*3}} + q_N^{{*5}} + q_N^{{*7}}) - 3 java.lang.StringIndexOutOfBoundsException: Range [0, 100) out of bounds for length 13
</tr>
<tr>
<td class="tdleft"><code class="code">4qN*3&5+&7</code></td>
<d class="tdcenter">means</td>
<td class="tdleft"><span class="SimpleMath">\(4 (q_N^{{*3}} + q_N^{{*5}}) + q_N^{{*7}}\)</span></td>
</tr>
</table><br />
</div>

<p>To explain the <q>ampersand</q> syntax in general we remark that <q>&k</q> is interpreted as <span java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 94


<class""pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "b7*3" );</span>
E(7)^3+E(7)^5+E(7)^6
<span class="GAPprompt">gap<iv class=func"java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 216
E(24)-E(24)^19
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "-3y'''24*13&5" );<<>Ifthe cyclotomic <var ="rg><var> is an irrational element of a quadratic extension of the rationals then <code class="func">StarCyc</code> returns the unique Galois conjugate of <var class="Arg">cyc</var> that is different from <var class="Arg">cyc</var>, otherwise <code class="keyw">fail</code> is returned. In the first case, the return value is often called <var class="Arg">cyc</var><span java.lang.StringIndexOutOfBoundsException: Index 413 out of bounds for length 0
3*E(8)-3*E(8)^3
<span class="GAPprompt">gap></span> /pre>/iv>
-3*E(24)-2*E(24java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
-3*(4)E(24^11+E(24)^7+*(24^19
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0
-7*E(24)-4*E(24)^11+4*E(24)<div class="unc"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Quadratic</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( function )</td></tr></table></div>
sjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 107
6*E(24)-6*E(24)^19
</pre></div>

<p><<p>If<var =">cyc<var>is not a cyclotomic integeror  not lie in a quadratic extension field of the rationals then <code class="keyw">fail</code> is returned.</p>

<h4>18.5 <span class="Heading">Galois Conjugacy of Cyclotomics</span></h4>

<p><a id="X79EE9097783128C4" namejava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0

<h5>18.5-1 GaloisCyc</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 26
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GaloisCyc</code>( <var class="Arg">list</var>, <var class="Arg">k</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a cyclotomic <var class="Arg">cyc</var> and an integer  =13b3"a =-  =3 d:2

<var class=""cyc</ <ode >GaloisCyc v java.lang.StringIndexOutOfBoundsException: Range [108, 107) out of bounds for length 303

<p>For a list or matrix <var class="Arg">list</var> of cyclotomics, <code class="func">GaloisCyc</code> returns the list obtained by applying <code class="func">GaloisCyc

<p><a id="X7BE001A0811CD599" name="X7BE001A0811CD599"></a></p>

<h5>18.5-2 ComplexConjugate</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ComplexConjugate</code>( <var class="Arg">z</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RealPart</code>( <var class="Arg">z</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="java.lang.StringIndexOutOfBoundsException: Range [0, 91) out of bounds for length 0
<p>For a cyclotomic number <var class="Arg">z</var>, <code class="func">>Let <var class="Arg">mat</var> be a matrix of cyclotomics. <code class="func">GaloisMatoperationof   the ( java.lang.StringIndexOutOfBoundsException: Range [197, 196) out of bounds for length 313

<p>When <code class="func">ComplexConjugate</code> is p>If some rows of <var class="Arg">mat</var> are identicalforthe a willbe <pjava.lang.StringIndexOutOfBoundsException: Index 150 out of bounds for length 150

<p>When <code class="func">ComplexConjugate</code> is defined for an object <var class="Arg">z</var> then <code class="func">RealPart</code> and <code class="func">ImaginaryPart</code> return


<div class="example"><pre>
<> <span class"GAPinput"GaloisCyc (5 +(5^4 2 )<
E(5)^2+E(5)^3
<span class="GAPprompt">gap></span> << java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 75
E()4
<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5) + E(5)^4, -1 );  # this value is real</span>
E(5)+E(5)^4
<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(15) + E(15)^4, 3 );</span>
E(5)+(5)4
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 94
E(7)^6
</pre></div>

<>a =X7E361C057E97CA66name=X7E361C057E97CA66">/>/>

<h5>18.5-3 StarCyc</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; StarCyc</code>( <var <dd>
<p>If the cyclotomic <var class="Arg">cyc</var> is an irrational element of a quadratic extension of the rationals then <code class="func">StarCyc</code> returns the unique Galois conjugate<>strong class"ark>code class=code"gjava.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 129


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">StarCyc( EB(5) ); StarCyc( E(5) );</span>
E(5)^2+E(5)^3
fail
</pre></div>

<p><a<dd>

<h5>18.5-4 Quadratic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Quadratic</code>( <var class="Arg">cyc</var> )</td><td java.lang.StringIndexOutOfBoundsException: Index 160 out of bounds for length 160
<p>Let <var class="Arg">cyc</var> be a cyclotomic integer that lies in a quadratic extension field of the rationals/

<p>If <var class="Arg">cyc</var> is not a cyclotomic integer or does not lie in a quadratic extension field of the rationals then <code class="keyw">fail</code> is returned.</p>

pIjava.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 527

<p>If <span class="SimpleMath">\(d = 1\)</span>, we have the possibilities <span class="SimpleMath">\(i_{{|n|}}\)</span> for <span class="SimpleMath">\(n < -1\)</span>, <span class="SimpleMath">\(a + b * i\)</span> for <span class="SimpleMath">\(n = -1\)</span>, <span class="SimpleMath">\(a + b * r_n\)</span> for <span class="SimpleMath">\(n > 0\)</span>. Furthermore if <span  generators :  12(,) (3(,)]


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Quadratic( EB(5) ); Quadratic( EB(27) );</span>
rec( ATLAS := "b5", a := -1, b := 1, d := 2,
  display := "(-1+Sqrt(5))/2", root := 5 )
rec( ATLAS := "1+3b3", a := -1, b := 3, d := 2,
  display := "(-1+3*Sqrt(-3))/2", root := -3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Quadratic(0); Quadratic( E(5) );</span>
rec( ATLAS := "0", a := 0, b := 0, d := 1, display := "0", root := 1 )
fail
</pre></div>

<p><a id="X7DDDEC3F80543B7D" name="X7DDDEC3F80543B7D"></a></p>

<h5>18.5-5 GaloisMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GaloisMat</code>( <var class="Arg">mat</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">mat</var> be a matrix of cyclotomics. <code class="func">GaloisMat</code> calculates the complete orbits under the operation of the Galois group of the (irrational) entries of <var class="Arg">mat</var>, and the permutations of rows corresponding to the generators of the Galois group.</p>

<p>If some rows of <var class="Arg">mat</var> are identical, only the first one is considered for the permutations, and a warning will be printed.</p>

<p><code class="func">GaloisMat</code> returns a record with the components <code class="code">mat</code>, <code class="code">galoisfams</code>, and <code class="code">generators</code>.</p>


<dl>
<dt><strong class="Mark"><code class="code">mat</code></strong></dt>
<dd><p>a list with initial segment being the rows of <var class="Arg">mat</var> (<em>not</em> shallow copies of these rows); the list consists of full orbits under the action of the Galois group of the entries of <var class="Arg">mat</var> defined above. The last rows in the list are those not contained in <var class="Arg">mat</var> but must be added in order to complete the orbits; so if the orbits were already complete, <var class="Arg">mat</var> and <code class="code">mat</code> have identical rows.</p>

</dd>
<dt><strong class="Mark"><code class="code">galoisfams</code></strong></dt>
<dd><p>a list that has the same length as the <code class="code">mat</code> component, its entries are either 1, 0, -1, or lists.</p>


<dl>
<dt><strong class="Mark"><code class="code">galoisfams[i] = 1</code></strong></dt>
<dd><p>means that <code class="code">mat[i]</code> consists of rationals, i.e., <code class="code">[ mat[i] ]</code> forms an orbit;</p>

</dd>
<dt><strong class="Mark"><code class="code">galoisfams[i] = -1</code></strong></dt>
<dd><p>means that <code class="code">mat[i]</code> contains unknowns (see Chapter <a href="chap74_mj.html#X7C1FAB6280A02CCB"><span class="RefLink">74</span></a>); in this case <code class="code">[ mat[i] ]</code> is regarded as an orbit, too, even if <code class="code">mat[i]</code> contains irrational entries;</p>

</dd>
<dt><strong class="Mark"><code class="code">galoisfams[i] = </code><span class="SimpleMath">\([ l_1, l_2 ]\)</span></strong></dt>
<dd><p>(a list) means that <code class="code">mat[i]</code> is the first element of its orbit in <code class="code">mat</code>, <span class="SimpleMath">\(l_1\)</span> is the list of positions of rows that form the orbit, and <span class="SimpleMath">\(l_2\)</span> is the list of corresponding Galois automorphisms (as exponents, not as functions); so we have <code class="code">mat</code><span class="SimpleMath">\([ l_1[j] ][k] = \)</span> <code class="code">GaloisCyc( mat</code><span class="SimpleMath">\([i][k], l_2[j]\)</span><code class="code"> )</code>;</p>

</dd>
<dt><strong class="Mark"><code class="code">galoisfams[i] = 0</code></strong></dt>
<dd><p>means that <code class="code">mat[i]</code> is an element of a nontrivial orbit but not the first element of it.</p>

</dd>
</dl>
</dd>
<dt><strong class="Mark"><code class="code">generators</code></strong></dt>
<dd><p>a list of permutations generating the permutation group corresponding to the action of the Galois group on the rows of <code class="code">mat</code>.</p>

</dd>
</dl>

<div class="example"><pre>
<GAPprompt">gap>/><class""GaloisMat([[E() (44 ] <span
rec( galoisfams := [ [ [ 1, 2, 3, 4 ], [ 1, 7, 5span class"GAPprompt>gt;/pan><class"APinput>([[1, , ,[1,E(), E(2]]);/java.lang.StringIndexOutOfBoundsException: Range [119, 118) out of bounds for length 119
  generators := [ (1,2)(3,4), (1,3)(2,4) ],
  mat := [ [ E(3), E(4) ], [ E(3), -E(4) ], [ E(3)^2, E(4) ],
      [ E(3)^2, -  generators := [(2,3) ,
<pan ><span><pan =G"GaloisMat[[ 1,1  , [1,(3,E)^2  ] )<s>
rec( galoisfams := [ 1, [ [ 2, 3 ], [ 1, 2 ] ], 0 ],
  generators := [ (2,3) ],
  mat := [ [ 1, 1, 1 </re>/div>
</pre></div>

<p><java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<h5>18.5-6 RationalizedMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RationalizedMat</code><iv class=func">class=fjava.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 225
<p>returns the list of rationalized rows of <var class="Arg">mat</var>, which must be a matrix of cyclotomics. This is the set of sums over orbits under the action of the Galois group of the entries of <var class="Arg">mat</var> (see <code class="func">GaloisMat</code> (<a href="chap18_mj.html#X7DDDEC3F80543B7D"><span class="RefLink">18.5-5</span></a>)), so the operation may be viewed as a kind of trace on the rows.</p>

<p>Note that no two rows of <var class="Arg">mat</var> should be equal.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:= [ [ 1, 1, 1 ], [ 1, E(3), E(3)^2 ], [ 1, E(3)^2, E(3) ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RationalizedMat( mat );</span>
[ [ 1, 1, 1 ], [ 2, -1, -1 ] ]
</pre></div>

<p><a id="X8557FC2D7ACD6105" name="X8557FC2D7ACD6105"></a></p>

<h4>18.6 <span class="Heading">Internally Represented Cyclotomics</span></h4>

java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 592

<p>The disadvantage of this approach is that the arithmetical operations are quite expensive, so the use of internally represented cyclotomics is not recommended for doing arithmetics over number fields, such as calculations with matrices of cyclotomics. But internally represented cyclotomics are good enough for dealing with irrationalities in character tables (see chapter <a href="chap71_mj.html#X7B7A9EE881E01C10"><span class="RefLink">71</span></a>).</p>

<p>For the representation of cyclotomics one has to recall that the   1 ,1] 1,E(3), E()^], [ 1,E(3 ]<span>

<p>A special integral basis of cyclotomic fields is chosen that allows one to easily convert arbitrary sums of roots of unity into the basis, as well as to convert a cyclotomic represented w.r.t. the basis into the smallest possible cyclotomic field. This basis is accessible in <strong class="pkg">GAP</strong>, see <a href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink"[ [1,1,  , [2,1,- ] ]

<p>Note that the set of all <span class="SimpleMath"<p>The implementation of  <em>nternally represented cyclotomic</em> is based on a list of length equal to its conductor. This means that the internal representation of a cyclotomic does <em>not</em> refer to the smallest number field but the smallest <em>cyclotomic</em> field containing it. The reason for this is the wish to reflect the natural embedding of two cyclotomic


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">E(5) * E(5)^2; ( E(5) + E(5)^4 ) * E(5)^2;</span>
E(3
E(5)+E(5)^3
<span class="GAPprompt">gap></span> <span class="GAPinput">( E(5) + E(5)^4 ) * E(5);</span>
-E(5)-E(5)^3-E(5)^4
</pre></div>

<p>An internally represented cyclotomic is always represented in the smallest cyclotomic field it is contained in. The internal coefficients list coincides

<>To avoid becomingunintentionallyvery ,orconsuming verylarge of   a  conductorofinternally  cyclotomics, by   toone million. canberaised(notlowered)using<class=func><code < =chap18_mj.#D3028777DE39709>span">86-1<>/> and accessed  <odeclass""><code < =chap18_mj.html#7D3028777DE39709>span class"RefLink".-</>/>.The maximum  of  limitis span class""\{8-\<spanon class=S"(\<>bit ,and  class""\2^32-)/  s ""\(/   So the   < ""<strong> is  reallythe <pan ="impleMath>\ℚ{b\</span./p>

<p>It should be emphasized that one disadvantage of representing<pan class=GAPprompt">gap&<span><pan class=GAPinput>5)*E5^;(E5)+E5)4)*E()2;span>

<p><a id="X7D3028777DE39709" name="X7D3028777DE39709"></a></p>

<h5>18.6-1 SetCyclotomicsLimit</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SetCyclotomicsLimit</code>( <var class="Arg">newlimit</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GetCyclotomicsLimit<pan class="">&;/pan span =GAPinput> (5  5^   E5)<span>
<p><code class="func">GetCyclotomicsLimit</code> returns the current limit on conductors of internally represented cyclotomic numbers</p>

d to increasethejava.lang.StringIndexOutOfBoundsException: Range [125, 124) out of bounds for length 405


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