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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 href="chap18_mj.html#X841C425281A6F775">18.1-3 IsCyclotomic</a></span>
<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><a href="chap18_mj.html#X7CBA6CB678E2B143">18.1-6 String</a></span>
<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>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X7808ECF37AA9004D">18.1-9 RoundCyc</a></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#X8712419182ECD8DD">18.1-14 IsGaussInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X7E6CF4947D0A56F7">18.1-15 IsGaussRat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X7FE3D5637B5485D0">18.1-16 DefaultField</a></span>
</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"><br /><span class="nocss"> </span><a href="chap18_mj.html#X8511B8DF83324C27">18.2-1 IsInfinity</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18_mj.html#X7F66A62384329705">18.3 <span class="Heading">Comparisons of Cyclotomics</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18_mj.html#X7B242083873DD74F">18.4 <span class="Heading">ATLAS Irrationalities</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X8414ED887AF36359">18.4-1 <span class="Heading">EB, EC, <span class="SimpleMath">\(\ldots\)</span>, EH</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X813CF4327C4B4D29">18.4-2 <span class="Heading">EI and ER</span></a>
</span>
<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></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X844F0EBF849EDEB3">18.4-5 NK</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X812E334E7A869D33">18.4-6 AtlasIrrationality</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18_mj.html#X79FE34337DF2CD10">18.5 <span class="Heading">Galois Conjugacy of Cyclotomics</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X79EE9097783128C4">18.5-1 GaloisCyc</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X7BE001A0811CD599">18.5-2 ComplexConjugate</a></span>
<span class="ContSS"><br /><span 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">18.5-4 Quadratic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X7DDDEC3F80543B7D">18.5-5 GaloisMat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X7BB9F5957AA8C082">18.5-6 RationalizedMat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18_mj.html#X8557FC2D7ACD6105">18.6 <span class="Heading">Internally Represented Cyclotomics</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18_mj.html#X7D3028777DE39709">18.6-1 SetCyclotomicsLimit</a></span>
</div></div>
</div>
<h3>18 <span class="Heading">Cyclotomic Numbers</span></h3>
<p><strong class="pkg">GAP</strong> admits computations in abelian extension fields of the rational number field <span class="SimpleMath">\(ℚ\)</span>, that is fields with abelian Galois group over <span class="SimpleMath">\(ℚ\)</span>. These fields are subfields of <em>cyclotomic fields</em> <span class="SimpleMath">\(ℚ(e_n)\)</span> where <span class="SimpleMath">\(e_n = \exp(2 \pi i/n)\)</span> is a primitive complex <span class="SimpleMath">\(n\)</span>-th root of unity. The elements of these fields are called <em>cyclotomics</em>.</p>
<p>Information concerning operations for domains of cyclotomics, for example certain integral bases of fields of cyclotomics, can be found in Chapter <a href="chap60_mj.html#X80510B5880521FDC"><span class="RefLink">60</span></a>. For more general operations that take a field extension as a –possibly optional– argument, e.g., <code class="func">Trace</code> (<a href="chap58_mj.html#X7DD17EB581200AD6"><span class="RefLink">58.3-5</span></a>) or <code class="func">Coefficients</code> (<a href="chap61_mj.html#X80B32F667BF6AFD8"><span class="RefLink">61.6-3</span></a>), see Chapter <a href="chap58_mj.html#X80A8E676814A19FD"><span class="RefLink">58</span></a>.</p>
<p><a id="X79E25C3085AA568F" name="X79E25C3085AA568F"></a></p>
<h4>18.1 <span class="Heading">Operations for Cyclotomics</span></h4>
<p><a id="X8631458886314588" name="X8631458886314588"></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 class="tdright">( operation )</td></tr></table></div>
<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(3)^2
-1/3*E(12)^7
</pre></div>
<p>A particular basis is used to express cyclotomics, see <a href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>; note that <code class="code">E(9)</code> is <em>not</em> a basis element, as the above example shows.</p>
<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 )</td></tr></table></div>
<p>is the domain of all cyclotomics.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">E(9) in Cyclotomics; 37 in Cyclotomics; 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 <span class="SimpleMath">\(z\)</span> defines a Galois automorphism if and only if <span class="SimpleMath">\(k\)</span> is coprime to the conductor (see <code class="func">Conductor</code> (<a href="chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>)) of <span class="SimpleMath">\(z\)</span>.</p>
<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 root of unity is a cyclotomic integer, character values are always cyclotomic integers, but all rationals which are not integers are not cyclotomic integers.</p>
<div class="example"><pre>
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Range [1, 27) out of bounds for length 7
java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 25
<span class="GAPprompt">gap></span> <span class="GAPinput">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 integer, ...</span>
1/2*E(5)-1/2*E(5)^2-1/2*E(5)^3+1/2*E(5)^4
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIntegralCyclotomic( r2 );</spanspan >br/><span ">nbsp;nbsp;/>href"html>.- RoundCyc</<spanjava.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
false
<span class="java.lang.StringIndexOutOfBoundsException: Range [53, 19) out of bounds for length 139
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="br/<span "ocss>nbsp <span< href"chap18_mjhtmlX7"18.1- sGaussRat</<span>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func<java.lang.StringIndexOutOfBoundsException: Range [77, 76) out of bounds for length 188
<p>The operation <code class="func">/
<ivclass""<>
<pan=GAPprompt"gapgt;</pan> span ="GAPinput>(E)1*5)^ )Int/*E(-32*(4))<span
E(5)
-()
</pre></div>
<p><a id="class="ContSSBlock
<h5>18.1-6 String</h5>
<class=unc"< class="width%>tr< "tdleft"< ="8227 </code> var class=""cyc<var> )java.lang.StringIndexOutOfBoundsException: Range [149, 148) out of bounds for length 213
<p>The operation <code class="func">String</code> returns for a cyclotomic <var class="Arg">cyc</var> a string corresponding to the way the cyclotomic is printed by <code class="func">ViewObj</code> (<a href="chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) and <code class="func">PrintObj</code> (<a href="chap6_mj.html#X815BF22186FD43C9"><span class="RefLink<span>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">String( E(5)+1/2*E(5)^2 ); String( 17/3 );</span>
<span
"/3"
/pre</iv>
p<a id"815D6EC57CBA9827 name=X815D6EC57CBA9827"<a<p>
h517 h5
<java.lang.StringIndexOutOfBoundsException: Range [219, 4) out of bounds for length 219
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Conductor</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For an element <var class="Arg">cyc</var> of a cyclotomic field, <code class="func"<span "ContSS"< /spanjava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 134
<div class="example"><pre>
spanclass"GAPprompt">&t<span span class"GAPinput"(0) (E)) ( E(2 ;/span>
1
5
12
</pre></div>
<"></a></p>
<h5>18.1-8 java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 7
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227
<p>returns<><strongclass="GAP/strong> admits computations in abelian extension fields of the rational number field <span class="SimpleMath">\(ℚ\)</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
3
/>>
<java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 62
<h5>18.1-9 RoundCyc</h5>
<div class="func"><table class="func" java.lang.StringIndexOutOfBoundsException: Range [0, 43) out of bounds for length 0
81>java.lang.StringIndexOutOfBoundsException: Range [167, 166) out of bounds for length 876
<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
<span class(3java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
(5E5^
-2*E(28)^3+E(28)^4-2*E(28)^11-2*E(28)^15java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
-2*E(28)^27
</pre></div>
<p><a id="X7AE2933985BE4C3E" name="X7AE2933985BE4C3E"></a></p>
<h5>18.1-10 CoeffsCycpisthe domainofallcyclotomics
<div class="func"><table class=class"APprompt"gap&<span spanclass="E()inCyclotomics 37 java.lang.StringIndexOutOfBoundsException: Range [101, 100) out of bounds for length 129
<> < "">cyc</> be cyclotomic conductor span=""(n\/ ((see class="func">Conductor</code> (<a href="chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>)). If <var class="Arg">N</var> is not a multiple of <span class="SimpleMath">\(n\)</span> then <code class="func">CoeffsCyc</code> returns <code class="keyw">fail</code> because <var class="Arg">cyc</var> cannot be expressed in terms of <var class="Arg">N</var>-th roots of unity. Otherwise <code class="func">CoeffsCyc</code> returns a list of length <var class="Arg">N</var> with entry at position <span class="SimpleMath">\(j\)</span> equal to the coefficient of <span class="SimpleMath">\(\exp(2 \pi i (j-1)/\textit{N})\)</span> if this root belongs to the <var class="Arg">N</var>-th Zumbroich basis (see <a href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>), and equal to zero otherwise. So we have <var class="Arg">cyc</var> = <code class="code">CoeffsCyc(</code> <var class="Arg">cyc</var>, <var class="Arg">N</var> <code class="code">) * List( [1..</code><var class="Arg">N</var><code class="code">], j -> E(</code><var class="Arg">N</var><code class="code">)^(j-1) )</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">cyc:= E(5)+E(5)^2;</span>
E(5)+E(5)^2
<span class="GAPprompt">gap></span> <span class="GAPinput">CoeffsCyc( cyc, 5 ); CoeffsCyc( cyc, 15 ); CoeffsCyc( cyc, 7 );</span>
[ 0, 1, 1, 0, 0 ]
[ 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, -1, 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 class="tdleft"><code class="func">&#an style='color: green'>8227; DenominatorCyc</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For 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">IsIntegralCyclotomic</code> (<a href="chap18_mj.html#X869750DA81EA0E67"><span class="RefLink">18.1-4</span></a>)). For rational numbers <var class="Arg">cyc</var>, the result is the same as that of <code class="func">DenominatorRat</code> (<a
<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>
<p>The external representation of a cyclotomic <var class="Arg">cyc</var> with conductor <span class="SimpleMath
"gt "java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 126
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtRepOfObj( E(5) ); CoeffsCyc( E(5), 5 );</span>
[,1 0,0 0]
[ 0, 1, 0, 0, 0 ]
<spanclass="GAPprompt">gap></span> <span class="GAPinput">CoeffsCyc( E(5), 15 );</span>
[0,0,0, 0 0,0 0, 0,-1,0 , 0,0 1 0 java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
</pre></div>
<p><a id="X7DDD51B983D5BC44" name=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>18.1-13 DescriptionOfRootOfUnity</h5>
<div class="func"><table div ="func">table class"" width100"<tr<td =java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 210
<p>Given a cyclotomic <var class="Arg">root</var> that is known to be a root of unity (this is <em>not</em> checked), <code class="func">DescriptionOfRootOfUnity</code> returns a list <span class="SimpleMath">\([ n, e ]\)</span> of coprime positive integers such that <var class="Arg">root</var> <span class="SimpleMath">\(=\)</span> <code classThe java.lang.StringIndexOutOfBoundsException: Range [18, 16) out of bounds for length 272
<div class="java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 4
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 109
-E(9)^4-E(9)^7
[ 9, 1 ]
h51816String
[ 6, 5 ]
java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 12
<<idX8712419182 ="8712419182java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
<5>.- /5>
<div class="func"><table class="func" width="100%"><tr><td java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 17
<p><code class="func<>a id=XD6"nameX815D6EC57CBA9827<a<p
<p><a id="X7E6CF4947D0A56F7" name="X7E6CF4947D0A56F7
<h5>18.1-15 IsGaussRat</h5>
<class"unc< =func" width"%>tr><td class="tdleft"><code class="func">‣ IsGaussRat</code>( <var class="Arg">x</var> )</td><td class="tdright">( function )< a field,<code java.lang.StringIndexOutOfBoundsException: Range [112, 79) out of bounds for length 650
<p">*E(4)</code>, where <span class="SimpleMath">\(a\)</span> and <span class="SimpleMath">\(b\)</span> are rationals.</p>
<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</var> )</td><td class="tdright">( function )</td></tr></table></div>
<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> (<a href="chap58_mj.html#X871AA7D58263E9AC"><span class="RefLink">58.1-3</span></a>) returns the smallest field containing all given elements, which need not be a cyclotomic field. In both cases, the fields represent vector spaces over the rationals (see <a href="chap60_mj.html#X7D2421AC8491D2BE"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<span ">gap>/span>< class=GAPinput"(E5+5^ ; DefaultField()+5^ )<java.lang.StringIndexOutOfBoundsException: Range [120, 119) out of bounds for length 120
NF, 1 4 )
CF(5)
</pre></div>
<< idX7EE5FB7181125E02="7E5FB7181125E02>/a><p>
<h4>18.2 <span class="java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 1
<p><a id="X8511B8DF83324C27" name="X8511B8DF83324C27"></a></p>
<h5>18.2-1 IsInfinity</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsInfinity</code>( <var <>a integer < classSimpleMath\z\)/>see<codeclass">IsIntegralCyclotomiccode>(< href="hap18_mjhtml#869750">span =RefLink>181-4/>a>) near the cyclotomic< ="Arg>cyc/> following:Let code java.lang.StringIndexOutOfBoundsException: Range [293, 292) out of bounds for length 876
<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="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; infinity</code></td><td class="tdright">( global variable )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; -infinity</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p><code class="func">infinity</code> and <code class="func">-infinity</code> are special <strong class="pkg">GAP</strong> objects that lie in <code class="code">CyclotomicsFamily</code>. They are larger or smaller than all other objects in this java.lang.StringIndexOutOfBoundsException: Index 252 out of bounds for length 0
<p>Some arithmetic operations are provided for convenience when using <code class="func">infinity</code> and <code class="func">-infinity</code> as top and bottom element respectively.</p>
<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
</spanclass="APprompt>&t;/> span =GAPinput"cyc=E5+()2<
<java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 551
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= Size( Rationals );</span>
infinity
<span class="GAPprompt">gap></span> <span class="GAPinput">s = infinity; IsCyclotomic,-,,0, 0, 0, 0, 0, -1, 0, 0, -1,0 -1,0]
true
true
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">s in Rationals; s &>java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 754
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( [ s, 2, s, E(17), s, 19 ] );</span>
[ 2, 19, E(17), infinity ]
</pre></div>
<p><a id="X7F66A62384329705" name="X7F66A62384329705"></a></p>
<h4>18.3 <span class="Heading">Comparisons of Cyclotomics</span></h4>
<p>To compare cyclotomics, the operators <code class="code"><</code>, <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="keyw">false
<p>Cyclotomics are ordered as follows: The relation between p>The external representation of a cyclotomic <var class="Arg">cyc<< class=SimpleMath"(\<span> (ee code ="<java.lang.StringIndexOutOfBoundsException: Range [168, 166) out of bounds for length 485
<p>For comparisons of cyclotomics and other <strong class="pkg">GAP</strong> objects, see Section <a href="chap4_mj.html#X7A274A1F8553B7E6"><span class="RefLink">4.13</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">E(5) < E(6); # the latter value has conductor 3</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">E(3) < E(3)^2; # both have conductor 3, compare the ext. repr.</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">3 < E(3); E(5) < E(7);</span>
true
true
</pre></div>
<p><a id="X7B242083873DD74F" name="X7B242083873DD74F"></a></p>
<>84 <spanclass=Heading"> Irrationalities<span></h4>
<p><a id="X8414ED887AF36359" name="X8414ED887AF36359"></a></p>
<>181 spanclass"Heading">EB,EC,<span class"impleMath"\\dots\)/, EH/span>/5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EB</code>( <var class="Arg">N</var> )<java.lang.StringIndexOutOfBoundsException: Index 0 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; EC[9,1 ]
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ED</code>( <var class="Arg">N</var> )</td><td class="tdright"[6,5]
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code<div class=="unc>< class=func"java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 217
<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< href"chap60_mj.#80"< class"">0.1<><a>) and class"">falsecode .Gaussianintegers are of <class"SimpleMath">\(a + b\)</span><code class="code">*E(4)</code>, where <span class="SimpleMath">\(a\)</span> and <span class="SimpleMath">\(b\)</span> are integers.</p>
=">class""width=100%>tr<d">=">#227;/ode( < "></ )/> ">&bsp;;<td</>/java.lang.StringIndexOutOfBoundsException: Range [203, 202) out of bounds for length 209
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EH</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></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
<div class="pcenter"><table class="GAPDocTablenoborder">
<tr>
<td class="tdleft"><code class="code">EB(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(b_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class"tdleft"><span class="SimpleMath">\(\left( \sum_{{j = 1}}^{{\textit{N}-1}} z^{{j^2}} \right) / 2\)</span></td>
<td class="tdcenter">,</td>
<td class="tdleft"><div ="example">prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
</tr>
<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 h4>18.2 class>and negative Infinity</span></h4>
<td class="tdleft"><span class="SimpleMath">\(\left( \sum_{{j = 1}java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<td class="tdcenter">,</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{3}\)</span></<h5>18.-1IsInfinity/h5>
</tr>
<tr>
<td class="tdleft"><code class="code">ED(</code><var class="Arg">N</var><code class="code">)</code></td>
<class="tdcenter">=</d
<td class="tdleft"><span class="SimpleMath">\(d_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<d =""< class=SimpleMath"\l(s{=1}}{\extit{N}1}z{j4} \)/4)<span>/>
<td class="tdcenter">,</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{4}\)</span></td>
</tr>
<tr>
<class=java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 104
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(e_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="
<td class="tdcenter">,</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{5}\
java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 5
<tr>
<td class="tdleft<iv class="example"><pre>
<td class="tdcenter">=</td>
<td "dleft"<panjava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 74
="</td
<td class="tdleft"><span class="SimpleMath">\(\leftjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<td class="tdcenter">,</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 4
</tr>
<tr>
<td class="tdleft"[2, 19 E(), infinity]
<td java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<td class="tdleft"><span class="SimpleMath">\(g_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(\left( \sum_{{j = 1<>To compare, operators codeclass=c">&/java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 461
"/>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{7}\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">EH(</code><var class="Arg">N</var><code class="code">)
< classtdcenter></d>
<td class="tdleft"><span class="SimpleMath">\(h_{\textit{N}}\)</span></td>
<ass="tdcenter">=</td
<td class="tdleft"><span class="SimpleMath">\(\left( \sum_{{j = 1}}
<td class="tdcenter">,</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod{8}\)</span></td>
</tr>
</table><< class=GAPprompt>ap;<span>span class="GAPinput"E3)& E3^; have conductor, the <>
/>
<p>(Note that in <code class/>/ivjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<div class="example"><pre>
<span class="GAPprompt"gapgt<span> <span class="GAPinput">EB(5); EB(9);</span>
E(5)+E(5)^4
1
/>/iv
<p><a id="X813CF4327C4B4D29" name="X813CF4327C4B4D29"></a></p>
<h5>18.4-2 <ivclass"unc><table class=func"=100%>trtd class"dleft"<class"unc>#8227;EB<code>( < class="rg"N/var> )/td>td class="tdright"(& )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EI</code>( <var class="Arg">Ndiv class=f"< classf"width="100%>tr< ="dleft"codeclass">#;ED</code> <ar class=rg>/var>)/d>td =>&function&java.lang.StringIndexOutOfBoundsException: Range [184, 183) out of bounds for length 209
<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>
<p>For a rational number <var class="Arg">N</var>, <code class="func">ER</code> returns the square root <span class="SimpleMath">\(\sqrt{{\textit{N}}}\)</span> of <var class="Arg">N</var>, and <code class="func">EI<div ="unc>table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EG</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr<iv class=func"<class=func"width"%>tr<tdclass"">code =""#8227EH<> var=Arg"N</var>)td<tdclass=t"(nbsp&;/d/tr<table<
<p><code class="func">ER</code> is installed as method for the operation <code class="func">Sqrt</code> (<a href="chap31_mj.html#X7E8F1FB87C229BB0"><span class="RefLink">31.12-5</span></a>), for rational argument.</p>
<p>From a theorem of Gauss (see <a href="chapBib_mj.html#biBLan70">[Lan70, QS4 in Ch. IV, Par. 3]</a>) we know that <span class="java.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 27
<=pcenter> "GAPDocTablenoborder">
<tr>
<td class="tdleft"><span class="SimpleMath">\((-1 + \sqrt{{\textit{N}}}) / 2\)</span></td>
<td class="tdcenter">if</td>
<td class="tdleft"><span class="SimpleMath">\(\textit{N} \equiv 1 \pmod 4\)</span></td>
</tr>
<tr>
<td /tr>
<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 computed from <span class="SimpleMath">\(b_{\textit{N}}\)</span>, see <code java.lang.StringIndexOutOfBoundsException: Index 150 out of bounds for length 27
<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
/>/>
<< ="=X8672>/a></p
<h5>18.4-3 <span class="Heading">EY, EX, <span class="SimpleMath">\(</r
<dclass="tdleft"><code class="code">EE(</code><var class="Arg">N</var><code class="code">)</code></td>
< class="func"><table class=func width="00%"<tr><dclass="tdleft"><code class="func">&#e='color: green'>8227; EX</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tabletdcenter></td>
<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<td =><tdjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code< java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 27
>java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 237
<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'>8227tr
<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<classt"=/
<p class="center">\[
1, -1, 2, -2, 3, -3< java.lang.StringIndexOutOfBoundsException: Range [11, 9) out of bounds for length 88
\]</p>
<p>e (with < class"impleMath"\z=\exp2\ii/\extit{}))/pan)</>
<<classtdleft>span=SimpleMath\\textit{}\/span</
<tr>
<td class="tdleft"><code class="code">EY(</code><="SimpleMath">\(\left( }{{{-}z{^}\java.lang.StringIndexOutOfBoundsException: Range [102, 101) out of bounds for length 120
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(y_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td /div>
<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 class="SimpleMath">\(x_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(z + z^n + z^{{n^2}<div class=""><pre
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_3)\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">EW</code>(<var class="Arg">N</var><code java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 12
<td class="tdcenter">=</td>
<td class="tdleft"><pan class="SimpleMath">\(w_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class2 <span class=Heading> and ER</span>/h5>
<td class="tdleft"><span class="SimpleMath">\<iv class="unc"<able""width"%"<>class="dleft"<codeclass="func"ã EI/code(< class="rg"</> )</d>tdclass="dright">&bsp; </td<<table><divjava.lang.StringIndexOutOfBoundsException: Index 209 out of bounds for length 209
</tr>
<tr>
<td class="tdleft"><code class="code">EV(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span<pFor a number "Arg"N<var>, code class="unc>ER</code returns the root <span class=="impleMath>\\sqrt{{}}<span of<ar java.lang.StringIndexOutOfBoundsException: Range [174, 173) out of bounds for length 819
<t"=/>
<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"
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(u_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="dleft"><span class="SimpleMath">\(z + z^n + z^{{n^2}} + \ldots + z^{{n^5}}\)</span></td>
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_6)\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">ET(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter< =t"< =SimpleMath>(textitNe14)/></td>
<td class"">spanclass"">({textitN}/>/>
<td class="td class="tdleftclass="(- +i\sqrt{textitN})/2<span>tdjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
<td class="tdleft"><span class="SimpleMath">\(z + z^n + z^{{n^2}} + \ldots + z^{{n^6}}\)</span><//java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 5
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_7)\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">ES(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(s_{\textit{N}}\)</span></td>
<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>
</tr>
</table><br />
</div>
<p>For the two-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>
java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 26
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
E(5)+E(5)^4
<span class="GAPprompt">gap></span> <span class="GAPinput">EW(16,3); EW(17,2);</span>
0
E(17)+E(17)^4+E(17)^13+E(17)^16
</pre></div>
<p><a idjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>18.4-4 <span class="Heading">EM, EL, <span class="SimpleMath">\(\ldots\)</span>, EJ</span></h5>
<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 class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr
<java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 104
<div class=class""<class=S">(\N}\</pan</td>
p<varA<java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 201
<div class="pcenter"><table class="GAPDocTablenoborder">
<tr>
<td class="tdleft"><code class="code">EM(</code><var class="Arg">N</var><code class="code">)</codet=>span""\( {java.lang.StringIndexOutOfBoundsException: Range [72, 60) out of bounds for length 78
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(m_{\textit{N}}\)</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">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">\(z - z^n + z^{{n^2}} - z^{{n^3}}\)</span></td>
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_4)\)</span></td>
</tr>
<tr>
<td class="tdleft"><code class="code">EK(</code><var class="Arg">N</var><code class="code">)</code></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(k_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(z - z^n + \ldots - z^{{n^5}}\)</span></td>
<td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_6)\)</span></td>
</tr>
<tr>
<td < java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 91
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(j_{\textit{N}}\)</span></td>
<td class="tdcenter">=</td>
<td class="tdleft"><span class="SimpleMath">\(z - z^n + \ldots - z^{{n^7}}\)</span></td>
td class="tdleft"><span class="SimpleMath">\((n = \textit{N}_8)\)</span></td>
</tr>
</table><r /
</div>
<p>For the two-argument versions of the functions, see Section <code class="func">NK</code> (<a href="chap18_mj.html#java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 104
<<a id="844F0EBF849EDEB3 name"X844F0EBF849EDEB3">/a><p>
java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 18
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NK<</tr>
<"java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 628
<p>We write <span class="SimpleMath">\(\textit{N}_{\textit{k}} = \textit{N}_{\textit{k}}^{(0)}, \textit{N}_{\textit{k}}^{\prime} = \textit{N}_{\textit{k}}^{(1)}, \textit{N}_{\textit{k}}^{\prime\prime} = \textit{N}_{\textit{k}}^{(2)}\)</span> and so on.</p>
<p>The algebraic numbers</p>
< ">[
y_{<var class="Arg">N</var>}^{\prime} = y_{<var class="Arg">N</var>}^{(1)},
y_{var class"><var>^\prime}=y_{ class"Arg"><var>^(2)}, \dotsjava.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
x_{<var class="< ""\s_{textitN})/<td>
j_{<var class="Arg">N</var>}^{\prime}, j_{<var class="Arg">N</var>}^{\java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 27
\]</p>
<p>are obtained on replacing <span class="SimpleMath">\(\textit{N}_{\textit{k}}\)</span> in the definitions<tr>
<div class="pcenterd>
< java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 181
<td class="tdleft"><code class="code">EY(</code><var class="Arg">N</var>
<td class="tdcenter">=</td>
<td class="dleft"<span class="impleMath>(_\extit{}}^({}})<span>/td>>
</tr>
<tr>
<td class="tdleft"><code class="code">EX(</code><var class="Arg">N</var>,<var class="Arg">d</var><code class="code">)<pan class="GAPprompt">gap></span> <span class="GAPinput">EW(16,3); EW(17,2);</span>
<class=tdcenter"<>
<td class="tdleft"><span class="SimpleMath">\(x_{\textit{N}}^{(\textit{d})}\)</span></td>
</tr>
<
<class=tdleft"<td>
<td class="tdcenter"><span class="SimpleMath">\(\ldots\)<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< class"java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 24
/
<djava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 237
<td class="tdleft"><code class="code">EJ(</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">\(j_{\textit{N}}^{(\textit{d})}\)</span></<p>Let <var class="Arg">N</var> be an integer, <var class="Arg">N</var> <span class="Sim(ith<pan class="">z =exp)</>)/>
</tr>
</table><br />
</div>
<p><<td class="dcenter"></d>
<5184-AtlasIrrationality</5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AtlasIrrationality</code>( <java.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 67
p><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</<
t
<><class=>(n \{}4\)span</td>
<td class="tdleft"><code class="code">2qN+3&5-4&7+&9</code></td>
<d class"tdcenter"means<td>
<class=""span class="SimpleMath">\(2 q_N + 3 q_N^{{*5}} - 4 q_N^{{*7}} + q_N^{{*9}}\)</span></td>
</tr>
<tr>
<td class=""><java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 79
<td class="tdcenter">means</td>
<td =java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 122
</tr>
<tr>
<td class="tdleft"><codetd class><td>
<td class="tdcenter">means</td>
<td class="tdleft"><span classclass=tdleft"><span class="SimpleMath">\(z - z^n + \ldots - z^{{n^7}}\)</span></td>
/
</table></
</div>
<To java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 796
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "b7*3" );</span>
E(7)^3+E(7)^5+E(7)^6
<span
E(24)-E(24)^19
<pan class"GAPprompt>&java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 111
3*8))-E()^3
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
-3E)-2*(24)^+2*E(24)+*(24)19
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "3y'''24*13-&5" );</span>
-3*E(24)-E(24)^11+E(24)^17+3*E(24)^19
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "3y'''24*13
-7*E(24)-4*E(24)^11+4*E(24)^17+7*E(24)^19
<pan =""gapgt</> <pan class="GAPinput">AtlasIrrationality( "3y'''24&7" );</span>
6*E24)6E(2419
</pre></div>
<p<aid"" name"X79FE34337DF2CD10"><><>
<h4>18.5 <span class="Heading">Galois Conjugacy of Cyclotomics</span></h4>
<< =X79EE9097783128C4" ="X79EE9097783128C4">a>/>
h>.5-GaloisCyc/
<div class="func"><table class="java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 4
<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 )</<java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 27
<p>For a cyclotomic <var class="Arg">cyc</var> and an integer java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 4
<p>The <em>complex =java.lang.StringIndexOutOfBoundsException: Range [21, 19) out of bounds for length 68
<p>For a list or matrix <var class="Arg">list</var> of cyclotomics, <code class="func">GaloisCyc</code> returns the list obtained by applying <td class="tdleft"><code class="code">EJ(</code><var class="Arg">N</var>,<var class="Arg">d</var><code class="code">)</code></td>
==>/>/java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<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; java.lang.StringIndexOutOfBoundsException: Index 111 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; 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="func">&#n style='color: green'>8227; ImaginaryPart</code>( <var class="Arg">z</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<<p =pcenter"><able =GAPDocTablenoborder
<p></code> is called java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 220
<p>When <code class="func">ComplexConjugate</code> is defined for an object <var class="Arg">z</var> then <code class="func">java.lang.StringIndexOutOfBoundsException: Index 131 out of bounds for length 5
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5) + E(5)^4, 2 );</span>
E(5)^2+E(5)^3
<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5), -1 ); # the complex conjugate</span>
E(5)^4
<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5) + E(5)^4, -1 ); < java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 31
E(5)+E(5)^4
<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(15) + E(15)^4, 3 );</span>
E(5)+E(5)^4
<span class="GAPprompt">gap></span> <span class="GAPinput">ComplexConjugate( E(7) );</span>
E(7)^6
</pre></div>
<p><a id=<iv class=example"<>
<h5>18.5-3 StarCyc</h5>
<"unc"<table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StarCyc</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( function )</td></tr></table></div>
p> cyclotomic< class=""cyc/java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 533
<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<d
<p><a id="X84438F867B0CC299" name="X84438F867B0CC299"></a></p>
<h53E()()1124^+*)19
="<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 218
<p>Let <var class="Arg">cyc</var> be a cyclotomic integer that lies in a quadratic extension field of the rationals. Then we have <var class="Arg">cyc</var><span class="SimpleMath">\( = (a + b \sqrt{{n}}) / d\)</span>, for integers <span class="SimpleMath">\(a\<pan class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "3y'''24&7" );</span>
classArg"/ acyclotomic doesjava.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 177
<p>If the denominator <span class="SimpleMath">\(d\)</span> is <span class="SimpleMath">\(2\)</java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 0
<p>If <span class="SimpleMath">\(d = 1\)</span>, we have the possibilities <span class="SimpleMath">\(i_{{|n|}}\)</span> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<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: "+3b3" a : -1, b: 3, : 2,
display := "(-1+3*Sqrt(-3))/2", root := -3 )
<span class="GAPprompt">gap></<p>The <em>complex conjugate</em> of "Arg>cyc</var>is <ode class="code"GaloisCyc( <ar class="Arg">cyc</var>, -1 )</code>, which can also be computed using <code class="func">ComplexConjugate</code> (<a href="chap18_mj.html#X7BE001A0811CD599"><span class="RefLink">18.5-2</span></a>).</p>
rec( ATLAS := "0", a := 0, b := 0, d := 1, display := "0", root := 1 )
</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</code> calculates the complete orbits under the operation of the Galoisgroup of irrational)entries of <var class="Arg">mat</var>, and the permutations of rows corresponding to the generators of the Galois group.</p>
<, only the first one is considered for the permutations, and warning will printed./>
<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 span class="GAPprompt">gap></span=GAPinput"(E)+E(^, 2)<span>
</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</(^java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
<dl>
<dtE5E5)^4
<dd<span class="GAPprompt">gap></span> <span class="GAPinput">ComplexConjugate( E(7) );</span>
</dd>
<dt><p<id"" name=X7E361C057E97CA66</<p
<dd><p>means that <code class="code">mat[i]</codejava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 23
<d>
<t><strong=""< class">aloisfams[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 | | |
