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<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 | |