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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (ref) - Chapter 18: Cyclotomic Numbers</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap18" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <p id="mathjaxlink" class="pcenter"><a href="chap18_mj.html">[MathJax on]</a></p> <p><a id="X7DFC03C187DE4841" name="X7DFC03C187DE4841"></a></p> <div class="ChapSects"><a 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class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X79EE9097783128C4"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7BE001A0811CD599"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7E361C057E97CA66"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X84438F867B0CC299"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7DDDEC3F80543B7D"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7BB9F5957AA8C082"> </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18.html#X </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7D3028777DE39709"> </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 ratio <p>Information concerning operations for domains of cyclotomics, for example certain integra <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">‣ E</ <p><code class="func">E</code> returns the primitive <var class="Arg">n</var>-th root of unity <span c <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.html#X7D2421AC8491D <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">‣ Cy <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 Cyclotomi true true false </pre></div> <p>As the cyclotomics are field elements, the usual arithmetic operators <code class="code">+</c <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(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">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Every object in the family <code class="code">CyclotomicsFamily</code> lies in the category <co <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomic(0); IsCyclotomic(1/ true true true <span class="GAPprompt">gap></span> <span class="GAPinput">IsCyc(0); IsCyc(1/2*E(3)); IsCyc( 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">‣ Is <p>A cyclotomic is called <em>integral</em> or a <em>cyclotomic integer</em> if all coefficients of its <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">r:= ER( 5 ); # The square root of 5 ...</s E(5)-E(5)^2-E(5)^3+E(5)^4 <span class="GAPprompt">gap></span> <span class="GAPinput">IsIntegralCyclotomic( r ); # ... is a true <span class="GAPprompt">gap></span> <span class="GAPinput">r2:= 1/2 * r; # This is not a cyclotomic 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 );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">r3:= 1/2 * r - 1/2; # ... but this is one.< 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> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ In <p>The operation <code class="func">Int</code> can be used to find a cyclotomic integer near to an ar <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Int( E(5)+1/2*E(5)^2 ); Int( 2/3*E( E(5) -E(4) </pre></div> <p><a id="X7CBA6CB678E2B143" name="X7CBA6CB678E2B143"></a></p> <h5>18.1-6 String</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ St <p>The operation <code class="func">String</code> returns for a cyclotomic <var class="Arg">cyc</v <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">String( E(5)+1/2*E(5)^2 ); String( "E(5)+1/2*E(5)^2" "17/3" </pre></div> <p><a id="X815D6EC57CBA9827" name="X815D6EC57CBA9827"></a></p> <h5>18.1-7 Conductor</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>For an element <var class="Arg">cyc</var> of a cyclotomic field, <code class="func">Conductor</c <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Conductor( 0 ); Conductor( E(10) ); C 1 5 12 </pre></div> <p><a id="X81DD58BB81FB3426" name="X81DD58BB81FB3426"></a></p> <h5>18.1-8 AbsoluteValue</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ab <p>returns the absolute value of a cyclotomic number <var class="Arg">cyc</var>. At the moment only <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">AbsoluteValue(-3);</span> 3 </pre></div> <p><a id="X7808ECF37AA9004D" name="X7808ECF37AA9004D"></a></p> <h5>18.1-9 RoundCyc</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ro <p>is a cyclotomic integer <span class="SimpleMath">z</span> (see <code class="func">IsIntegralC <p>Expressed in terms of the Zumbroich basis (see <a href="chap60.html#X7D2421AC8491D2BE"><spa <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">RoundCyc( E(5)+1/2*E(5)^2 ); Round 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">‣ Co <p>Let <var class="Arg">cyc</var> be a cyclotomic with conductor <span class="SimpleMath">n</span> ( <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, [ 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">‣ De <p>For a cyclotomic number <var class="Arg">cyc</var> (see <code class="func">IsCyclotomic</code> (< <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">‣ Ex <p>The external representation of a cyclotomic <var class="Arg">cyc</var> with conductor <span cla <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">ExtRepOfObj( E(5) ); CoeffsCyc( E(5 [ 0, 1, 0, 0, 0 ] [ 0, 1, 0, 0, 0 ] <span class="GAPprompt">gap></span> <span class="GAPinput">CoeffsCyc( E(5), 15 );</span> [ 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0 ] </pre></div> <p><a id="X7DDD51B983D5BC44" name="X7DDD51B983D5BC44"></a></p> <h5>18.1-13 DescriptionOfRootOfUnity</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ De <p>Given a cyclotomic <var class="Arg">root</var> that is known to be a root of unity (this is <em>not</e <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">E(9); DescriptionOfRootOfUnity( E -E(9)^4-E(9)^7 [ 9, 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">DescriptionOfRootOfUnity( -E(3) ) [ 6, 5 ] </pre></div> <p><a id="X8712419182ECD8DD" name="X8712419182ECD8DD"></a></p> <h5>18.1-14 IsGaussInt</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p><code class="func">IsGaussInt</code> returns <code class="keyw">true</code> if the object <var cl <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">‣ Is <p><code class="func">IsGaussRat</code> returns <code class="keyw">true</code> if the object <var cl <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">‣ De <p><code class="func">DefaultField</code> for cyclotomics is defined to return the smallest <em>cy <p>Note that <code class="func">Field</code> (<a href="chap58.html#X871AA7D58263E9AC"><span cla <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Field( E(5)+E(5)^4 ); DefaultField NF(5,[ 1, 4 ]) CF(5) </pre></div> <p><a id="X7EE5FB7181125E02" name="X7EE5FB7181125E02"></a></p> <h4>18.2 <span class="Heading">Infinity and negative Infinity</span></h4> <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">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ in <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ -i <p><code class="func">infinity</code> and <code class="func">-infinity</code> are special <strong c <p>Some arithmetic operations are provided for convenience when using <code class="func">infin <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</q> cyclotomi <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( s ); IsCy true true false true <span class="GAPprompt">gap></span> <span class="GAPinput">s in Rationals; s > 17;</span> 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>, <c <p>Cyclotomics are ordered as follows: The relation between rationals is the natural one, ratio <p>For comparisons of cyclotomics and other <strong class="pkg">GAP</strong> objects, see Sectio <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">E(5) < E(6); # the latter value has cond false <span class="GAPprompt">gap></span> <span class="GAPinput">E(3) < E(3)^2; # both have conductor 3, 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> <h4>18.4 <span class="Heading">ATLAS Irrationalities</span></h4> <p><a id="X8414ED887AF36359" name="X8414ED887AF36359"></a></p> <h5>18.4-1 <span class="Heading">EB, EC, <span class="SimpleMath">...</span>, EH</span></h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EB< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EC< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ED< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EE< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EF< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EG< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EH< <p>For a positive integer <var class="Arg">N</var>, let <span class="SimpleMath">z =</span> <code clas <div class="pcenter"><table class="GAPDocTablenoborder"> <tr> <td class="tdleft"><code class="code">EB(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">b_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">( ∑_{j = 1}^{<var class="Arg">N</var>-1} z^{j^2} ) / 2</ <td class="tdcenter">,</td> <td class="tdleft"><span class="SimpleMath"><var class="Arg">N</var> ≡ 1 mod 2</span></td> </tr> <tr> <td class="tdleft"><code class="code">EC(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">c_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">( ∑_{j = 1}^{<var class="Arg">N</var>-1} z^{j^3} ) / 3</ <td class="tdcenter">,</td> <td class="tdleft"><span class="SimpleMath"><var class="Arg">N</var> ≡ 1 mod 3</span></td> </tr> <tr> <td class="tdleft"><code class="code">ED(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">d_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">( ∑_{j = 1}^{<var class="Arg">N</var>-1} z^{j^4} ) / 4</ <td class="tdcenter">,</td> <td class="tdleft"><span class="SimpleMath"><var class="Arg">N</var> ≡ 1 mod 4</span></td> </tr> <tr> <td class="tdleft"><code class="code">EE(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">e_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">( ∑_{j = 1}^{<var class="Arg">N</var>-1} z^{j^5} ) / 5</ <td class="tdcenter">,</td> <td class="tdleft"><span class="SimpleMath"><var class="Arg">N</var> ≡ 1 mod 5</span></td> </tr> <tr> <td class="tdleft"><code class="code">EF(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">f_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">( ∑_{j = 1}^{<var class="Arg">N</var>-1} z^{j^6} ) / 6</ <td class="tdcenter">,</td> <td class="tdleft"><span class="SimpleMath"><var class="Arg">N</var> ≡ 1 mod 6</span></td> </tr> <tr> <td class="tdleft"><code class="code">EG(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">g_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">( ∑_{j = 1}^{<var class="Arg">N</var>-1} z^{j^7} ) / 7</ <td class="tdcenter">,</td> <td class="tdleft"><span class="SimpleMath"><var class="Arg">N</var> ≡ 1 mod 7</span></td> </tr> <tr> <td class="tdleft"><code class="code">EH(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">h_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">( ∑_{j = 1}^{<var class="Arg">N</var>-1} z^{j^8} ) / 8</ <td class="tdcenter">,</td> <td class="tdleft"><span class="SimpleMath"><var class="Arg">N</var> ≡ 1 mod 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>, <sp <div class="example"><pre> <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> <h5>18.4-2 <span class="Heading">EI and ER</span></h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EI< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ER< <p>For a rational number <var class="Arg">N</var>, <code class="func">ER</code> returns the square ro <p><code class="func">ER</code> is installed as method for the operation <code class="func">Sqrt</c <p>From a theorem of Gauss (see <a href="chapBib.html#biBLan70">[Lan70, QS4 in Ch. IV, Par. 3]</a>) <div class="pcenter"><table class="GAPDocTablenoborder"> <tr> <td class="tdleft"><span class="SimpleMath">(-1 + sqrt{<var class="Arg">N</var>}) / 2</span></td> <td class="tdcenter">if</td> <td class="tdleft"><span class="SimpleMath"><var class="Arg">N</var> ≡ 1 mod 4</span></td> </tr> <tr> <td class="tdleft"><span class="SimpleMath">(-1 + i sqrt{<var class="Arg">N</var>}) / 2</span></td> <td class="tdcenter">if</td> <td class="tdleft"><span class="SimpleMath"><var class="Arg">N</var> ≡ -1 mod 4</span></td> </tr> </table><br /> </div> <p>So <span class="SimpleMath">sqrt{<var class="Arg">N</var>}</span> can be computed from <span clas <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">...</span>, ES</span></h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EY< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EX< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EW< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EV< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EU< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ET< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ES< <p>For the given integer <var class="Arg">N</var> <span class="SimpleMath">> 2</span>, let <span class= <p class="pcenter">1, -1, 2, -2, 3, -3, 4, -4, ... .</p> <p>We define (with <span class="SimpleMath">z = exp(2 π i/<var class="Arg">N</var>)</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">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">y_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">z + z^n</span></td> <td class="tdleft"><span class="SimpleMath">(n = <var class="Arg">N</var>_2)</span></td> </tr> <tr> <td class="tdleft"><code class="code">EX(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">x_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">z + z^n + z^{n^2}</span></td> <td class="tdleft"><span class="SimpleMath">(n = <var class="Arg">N</var>_3)</span></td> </tr> <tr> <td class="tdleft"><code class="code">EW</code>(<var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">w_<var class="Arg">N</var></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 = <var class="Arg">N</var>_4)</span></td> </tr> <tr> <td class="tdleft"><code class="code">EV(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">v_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <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 = <var class="Arg">N</var>_5)</span></td> </tr> <tr> <td class="tdleft"><code class="code">EU(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">u_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">z + z^n + z^{n^2} + ... + z^{n^5}</span></td> <td class="tdleft"><span class="SimpleMath">(n = <var class="Arg">N</var>_6)</span></td> </tr> <tr> <td class="tdleft"><code class="code">ET(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">t_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">z + z^n + z^{n^2} + ... + z^{n^6}</span></td> <td class="tdleft"><span class="SimpleMath">(n = <var class="Arg">N</var>_7)</span></td> </tr> <tr> <td class="tdleft"><code class="code">ES(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">s_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">z + z^n + z^{n^2} + ... + z^{n^7}</span></td> <td class="tdleft"><span class="SimpleMath">(n = <var class="Arg">N</var>_8)</span></td> </tr> </table><br /> </div> <p>For the two-argument versions of the functions, see Section <code class="func">NK</code> (<a hre <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">EY(5);</span> 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 id="X7E5985FC846C5201" name="X7E5985FC846C5201"></a></p> <h5>18.4-4 <span class="Heading">EM, EL, <span class="SimpleMath">...</span>, EJ</span></h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EM< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EL< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EK< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EJ< <p>Let <var class="Arg">N</var> be an integer, <var class="Arg">N</var> <span class="SimpleMath">> 2</sp <div class="pcenter"><table class="GAPDocTablenoborder"> <tr> <td class="tdleft"><code class="code">EM(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">m_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">z - z^n</span></td> <td class="tdleft"><span class="SimpleMath">(n = <var class="Arg">N</var>_2)</span></td> </tr> <tr> <td class="tdleft"><code class="code">EL(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">l_<var class="Arg">N</var></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 = <var class="Arg">N</var>_4)</span></td> </tr> <tr> <td class="tdleft"><code class="code">EK(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">k_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">z - z^n + ... - z^{n^5}</span></td> <td class="tdleft"><span class="SimpleMath">(n = <var class="Arg">N</var>_6)</span></td> </tr> <tr> <td class="tdleft"><code class="code">EJ(</code><var class="Arg">N</var><code class="code">)</cod <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">j_<var class="Arg">N</var></span></td> <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">z - z^n + ... - z^{n^7}</span></td> <td class="tdleft"><span class="SimpleMath">(n = <var class="Arg">N</var>_8)</span></td> </tr> </table><br /> </div> <p>For the two-argument versions of the functions, see Section <code class="func">NK</code> (<a hre <p><a id="X844F0EBF849EDEB3" name="X844F0EBF849EDEB3"></a></p> <h5>18.4-5 NK</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NK< <p>Let <span class="SimpleMath"><var class="Arg">N</var>_<var class="Arg">k</var>^(<var class="Arg <p>We write <span class="SimpleMath"><var class="Arg">N</var>_<var class="Arg">k</var> = <var class=" <p>The algebraic numbers</p> <p class="pcenter">y_<var class="Arg">N</var>^' = y_N^(1), y_N^'' = y_N^(2), ..., x_N^', <p>are obtained on replacing <span class="SimpleMath"><var class="Arg">N</var>_<var class="Arg">k< <div class="pcenter"><table class="GAPDocTablenoborder"> <tr> <td class="tdleft"><code class="code">EY(</code><var class="Arg">N</var>,<var class="Arg">d</var><c <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">y_<var class="Arg">N</var>^(<var class="Arg">d</var> </tr> <tr> <td class="tdleft"><code class="code">EX(</code><var class="Arg">N</var>,<var class="Arg">d</var><c <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">x_<var class="Arg">N</var>^(<var class="Arg">d</var> </tr> <tr> <td class="tdleft"></td> <td class="tdcenter"><span class="SimpleMath">...</span></td> <td class="tdleft"></td> </tr> <tr> <td class="tdleft"><code class="code">EJ(</code><var class="Arg">N</var>,<var class="Arg">d</var><c <td class="tdcenter">=</td> <td class="tdleft"><span class="SimpleMath">j_<var class="Arg">N</var>^(<var class="Arg">d</var> </tr> </table><br /> </div> <p><a id="X812E334E7A869D33" name="X812E334E7A869D33"></a></p> <h5>18.4-6 AtlasIrrationality</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ At <p>Let <var class="Arg">irratname</var> be a string that describes an irrational value as a linear c <div class="pcenter"><table class="GAPDocTablenoborder"> <tr> <td class="tdleft"><code class="code">2qN+3&5-4&7+&9</code></td> <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 q_N^{*1 </tr> <tr> <td class="tdleft"><code class="code">4qN*3&5+&7</code></td> <td 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 class <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "b7*3" );</spa E(7)^3+E(7)^5+E(7)^6 <span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "y'''24" );</s E(24)-E(24)^19 <span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "-3y'''24*13& 3*E(8)-3*E(8)^3 <span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "3y'''24*13- -3*E(24)-2*E(24)^11+2*E(24)^17+3*E(24)^19 <span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "3y'''24*13-& -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 <span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "3y'''24&7" );< 6*E(24)-6*E(24)^19 </pre></div> <p><a id="X79FE34337DF2CD10" name="X79FE34337DF2CD10"></a></p> <h4>18.5 <span class="Heading">Galois Conjugacy of Cyclotomics</span></h4> <p><a id="X79EE9097783128C4" name="X79EE9097783128C4"></a></p> <h5>18.5-1 GaloisCyc</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ga <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ga <p>For a cyclotomic <var class="Arg">cyc</var> and an integer <var class="Arg">k</var>, <code class="f <p>The <em>complex conjugate</em> of <var class="Arg">cyc</var> is <code class="code">GaloisCyc( <var c |
