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<div class="ChapSects"><a href="chap60.html#X80510B5880521FDC">60 <span class="Heading">Abelian Number Fields</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60.html#X7D4E43E5799753B5">60.1 <span class="Heading">Construction of Abelian Number Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X80D21D80850EFA4B">60.1-1 CyclotomicField</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X80E5AD028143E11E">60.1-2 AbelianNumberField</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X82F53C65802FF551">60.1-3 GaussianRationals</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60.html#X81B5FE06781DB824">60.2 <span class="Heading">Operations for Abelian Number Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X7B0AB0FB7A4136C4">60.2-1 Factors</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X87D78F5E875F2E8A">60.2-2 IsNumberField</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X7D202D707D5708FA">60.2-3 IsAbelianNumberField</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X84CAE4627F0CD639">60.2-4 IsCyclotomicField</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X87E7313D8070B9CC">60.2-5 GaloisStabilizer</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60.html#X7D2421AC8491D2BE">60.3 <span class="Heading">Integral Bases of Abelian Number Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X7F52BEA0862E06F2">60.3-1 ZumbroichBase</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X87DB9C2C858B722A">60.3-2 LenstraBase</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60.html#X7E4AB4B17C7BA10C">60.4 <span class="Heading">Galois Groups of Abelian Number Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X7B55A90582E818F3">60.4-1 GaloisGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X8643D4B47A827D9D">60.4-2 ANFAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60.html#X85E9E90D7FE877CC">60.5 <span class="Heading">Gaussians</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X80BD5EAB879F096E">60.5-1 GaussianIntegers</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60.html#X7BFD33D27BFB7C5A">60.5-2 IsGaussianIntegers</a></span>
</div></div>
</div>
<h3>60 <span class="Heading">Abelian Number Fields</span></h3>
<p>An <em>abelian number field</em> is a field in characteristic zero that is a finite dimensional normal extension of its prime field such that the Galois group is abelian. In <strong class="pkg">GAP</strong>, one implementation of abelian number fields is given by fields of cyclotomic numbers (see Chapter <a href="chap18.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>). Note that abelian number fields can also be constructed with the more general <code class="func">AlgebraicExtension</code> (<a href="chap67.html#X7CDA90537D2BAC8A"><span class="RefLink">67.1-1</span></a>), a discussion of advantages and disadvantages can be found in <a href="chap18.html#X8557FC2D7ACD6105"><span class="RefLink">18.6</span></a>. The functions described in this chapter have been developed for fields whose elements are in the filter <code class="func">IsCyclotomic</code> (<a href="chap18.html#X841C425281A6F775"><span class="RefLink">18.1-3</span></a>), they may or may not work well for abelian number fields consisting of other kinds of elements.</p>
<p>Throughout this chapter, <span class="SimpleMath">ℚ_n</span> will denote the cyclotomic field generated by the field <span class="SimpleMath">ℚ</span> of rationals together with <span class="SimpleMath">n</span>-th roots of unity.</p>
<p>In <a href="chap60.html#X7D4E43E5799753B5"><span class="RefLink">60.1</span></a>, constructors for abelian number fields are described, <a href="chap60.html#X81B5FE06781DB824"><span class="RefLink">60.2</span></a> introduces operations for abelian number fields, <a href="chap60.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a> deals with the vector space structure of abelian number fields, and <a href="chap60.html#X7E4AB4B17C7BA10C"><span class="RefLink">60.4</span></a> describes field automorphisms of abelian number fields,</p>
<p><a id="X7D4E43E5799753B5" name="X7D4E43E5799753B5"></a></p>
<h4>60.1 <span class="Heading">Construction of Abelian Number Fields</span></h4>
<p>Besides the usual construction using <code class="func">Field</code> (<a href="chap58.html#X871AA7D58263E9AC"><span class="RefLink">58.1-3</span></a>) or <code class="func">DefaultField</code> (<a href="chap18.html#X7FE3D5637B5485D0"><span class="RefLink">18.1-16</span></a>) (see <code class="func">DefaultField</code> (<a href="chap18.html#X7FE3D5637B5485D0"><span class="RefLink">18.1-16</span></a>)), abelian number fields consisting of cyclotomics can be created with <code class="func">CyclotomicField</code> (<a href="chap60.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) and <code class="func">AbelianNumberField</code> (<a href="chap60.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>).</p>
<p><a id="X80D21D80850EFA4B" name="X80D21D80850EFA4B"></a></p>
<h5>60.1-1 CyclotomicField</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclotomicField</code>( [<var class="Arg">subfield</var>, ]<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">‣ CyclotomicField</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">gens</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">‣ CF</code>( [<var class="Arg">subfield</var>, ]<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">‣ CF</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first version creates the <var class="Arg">n</var>-th cyclotomic field <span class="SimpleMath">ℚ_n</span>. The second version creates the smallest cyclotomic field containing the elements in the list <var class="Arg">gens</var>. In both cases the field can be generated as an extension of a designated subfield <var class="Arg">subfield</var> (cf. <a href="chap60.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p>
<p><code class="func">CyclotomicField</code> can be abbreviated to <code class="func">CF</code>, this form is used also when <strong class="pkg">GAP</strong> prints cyclotomic fields.</p>
<p>Fields constructed with the one argument version of <code class="func">CF</code> are stored in the global list <code class="code">CYCLOTOMIC_FIELDS</code>, so repeated calls of <code class="func">CF</code> just fetch these field objects after they have been created once.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclotomicField( 5 ); CyclotomicField( [ Sqrt(3) ] );</span>
CF(5)
CF(12)
<span class="GAPprompt">gap></span> <span class="GAPinput">CF( CF(3), 12 ); CF( CF(4), [ Sqrt(7) ] );</span>
AsField( CF(3), CF(12) )
AsField( GaussianRationals, CF(28) )
</pre></div>
<p><a id="X80E5AD028143E11E" name="X80E5AD028143E11E"></a></p>
<h5>60.1-2 AbelianNumberField</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianNumberField</code>( <var class="Arg">n</var>, <var class="Arg">stab</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">‣ NF</code>( <var class="Arg">n</var>, <var class="Arg">stab</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a positive integer <var class="Arg">n</var> and a list <var class="Arg">stab</var> of prime residues modulo <var class="Arg">n</var>, <code class="func">AbelianNumberField</code> returns the fixed field of the group described by <var class="Arg">stab</var> (cf. <code class="func">GaloisStabilizer</code> (<a href="chap60.html#X87E7313D8070B9CC"><span class="RefLink">60.2-5</span></a>)), in the <var class="Arg">n</var>-th cyclotomic field. <code class="func">AbelianNumberField</code> is mainly thought for internal use and for printing fields in a standard way; <code class="func">Field</code> (<a href="chap58.html#X871AA7D58263E9AC"><span class="RefLink">58.1-3</span></a>) (cf. also <a href="chap60.html#X81B5FE06781DB824"><span class="RefLink">60.2</span></a>) is probably more suitable if one knows generators of the field in question.</p>
<p><code class="func">AbelianNumberField</code> can be abbreviated to <code class="func">NF</code>, this form is used also when <strong class="pkg">GAP</strong> prints abelian number fields.</p>
<p>Fields constructed with <code class="func">NF</code> are stored in the global list <code class="code">ABELIAN_NUMBER_FIELDS</code>, so repeated calls of <code class="func">NF</code> just fetch these field objects after they have been created once.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NF( 7, [ 1 ] );</span>
CF(7)
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= NF( 7, [ 1, 2 ] ); Sqrt(-7); Sqrt(-7) in f;</span>
NF(7,[ 1, 2, 4 ])
E(7)+E(7)^2-E(7)^3+E(7)^4-E(7)^5-E(7)^6
true
</pre></div>
<p><a id="X82F53C65802FF551" name="X82F53C65802FF551"></a></p>
<h5>60.1-3 GaussianRationals</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GaussianRationals</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">‣ IsGaussianRationals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p><code class="func">GaussianRationals</code> is the field <span class="SimpleMath">ℚ_4 = ℚ(sqrt{-1})</span> of Gaussian rationals, as a set of cyclotomic numbers, see Chapter <a href="chap18.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a> for basic operations. This field can also be obtained as <code class="code">CF(4)</code> (see <code class="func">CyclotomicField</code> (<a href="chap60.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>)).</p>
<p>The filter <code class="func">IsGaussianRationals</code> returns <code class="keyw">true</code> for the <strong class="pkg">GAP</strong> object <code class="func">GaussianRationals</code>, and <code class="keyw">false</code> for all other <strong class="pkg">GAP</strong> objects.</p>
<p>(For details about the field of rationals, see Chapter <code class="func">Rationals</code> (<a href="chap17.html#X7B6029D18570C08A"><span class="RefLink">17.1-1</span></a>).)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CF(4) = GaussianRationals;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Sqrt(-1) in GaussianRationals;</span>
true
</pre></div>
<p><a id="X81B5FE06781DB824" name="X81B5FE06781DB824"></a></p>
<h4>60.2 <span class="Heading">Operations for Abelian Number Fields</span></h4>
<p>For operations for elements of abelian number fields, e.g., <code class="func">Conductor</code> (<a href="chap18.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>) or <code class="func">ComplexConjugate</code> (<a href="chap18.html#X7BE001A0811CD599"><span class="RefLink">18.5-2</span></a>), see Chapter <a href="chap18.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>.</p>
<p><a id="X7B0AB0FB7A4136C4" name="X7B0AB0FB7A4136C4"></a></p>
<h5>60.2-1 Factors</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Factors</code>( <var class="Arg">F</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Factoring of polynomials over abelian number fields consisting of cyclotomics works in principle but is not very efficient if the degree of the field extension is large.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:= Indeterminate( CF(5) );</span>
x_1
<span class="GAPprompt">gap></span> <span class="GAPinput">Factors( PolynomialRing( Rationals ), x^5-1 );</span>
[ x_1-1, x_1^4+x_1^3+x_1^2+x_1+1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Factors( PolynomialRing( CF(5) ), x^5-1 );</span>
[ x_1-1, x_1+(-E(5)), x_1+(-E(5)^2), x_1+(-E(5)^3), x_1+(-E(5)^4) ]
</pre></div>
<p><a id="X87D78F5E875F2E8A" name="X87D78F5E875F2E8A"></a></p>
<h5>60.2-2 IsNumberField</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNumberField</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a finite dimensional extension of a prime field in characteristic zero, and <code class="keyw">false</code> otherwise.</p>
<p><a id="X7D202D707D5708FA" name="X7D202D707D5708FA"></a></p>
<h5>60.2-3 IsAbelianNumberField</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAbelianNumberField</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a number field (see <code class="func">IsNumberField</code> (<a href="chap60.html#X87D78F5E875F2E8A"><span class="RefLink">60.2-2</span></a>)) that is a Galois extension of the prime field, with abelian Galois group (see <code class="func">GaloisGroup</code> (<a href="chap58.html#X80CAA5BA82F09ED2"><span class="RefLink">58.3-1</span></a>)).</p>
<p><a id="X84CAE4627F0CD639" name="X84CAE4627F0CD639"></a></p>
<h5>60.2-4 IsCyclotomicField</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCyclotomicField</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a <em>cyclotomic field</em>, i.e., an abelian number field (see <code class="func">IsAbelianNumberField</code> (<a href="chap60.html#X7D202D707D5708FA"><span class="RefLink">60.2-3</span></a>)) that can be generated by roots of unity.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNumberField( CF(9) ); IsAbelianNumberField( Field( [ ER(3) ] ) );</span>
true
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNumberField( GF(2) );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( CF(9) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( Field( [ Sqrt(-3) ] ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( Field( [ Sqrt(3) ] ) );</span>
false
</pre></div>
<p><a id="X87E7313D8070B9CC" name="X87E7313D8070B9CC"></a></p>
<h5>60.2-5 GaloisStabilizer</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GaloisStabilizer</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">F</var> be an abelian number field (see <code class="func">IsAbelianNumberField</code> (<a href="chap60.html#X7D202D707D5708FA"><span class="RefLink">60.2-3</span></a>)) with conductor <span class="SimpleMath">n</span>, say. (This means that the <span class="SimpleMath">n</span>-th cyclotomic field is the smallest cyclotomic field containing <var class="Arg">F</var>, see <code class="func">Conductor</code> (<a href="chap18.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>).) <code class="func">GaloisStabilizer</code> returns the set of all those integers <span class="SimpleMath">k</span> in the range <span class="SimpleMath">[ 1 .. n ]</span> such that the field automorphism induced by raising <span class="SimpleMath">n</span>-th roots of unity to the <span class="SimpleMath">k</span>-th power acts trivially on <var class="Arg">F</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r5:= Sqrt(5);</span>
E(5)-E(5)^2-E(5)^3+E(5)^4
<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( r5, 4 ) = r5; GaloisCyc( r5, 2 ) = r5;</span>
true
false
<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisStabilizer( Field( [ r5 ] ) );</span>
[ 1, 4 ]
</pre></div>
<p><a id="X7D2421AC8491D2BE" name="X7D2421AC8491D2BE"></a></p>
<h4>60.3 <span class="Heading">Integral Bases of Abelian Number Fields</span></h4>
<p>Each abelian number field is naturally a vector space over <span class="SimpleMath">ℚ</span>. Moreover, if the abelian number field <span class="SimpleMath">F</span> contains the <span class="SimpleMath">n</span>-th cyclotomic field <span class="SimpleMath">ℚ_n</span> then <span class="SimpleMath">F</span> is a vector space over <span class="SimpleMath">ℚ_n</span>. In <strong class="pkg">GAP</strong>, each field object represents a vector space object over a certain subfield <span class="SimpleMath">S</span>, which depends on the way <span class="SimpleMath">F</span> was constructed. The subfield <span class="SimpleMath">S</span> can be accessed as the value of the attribute <code class="func">LeftActingDomain</code> (<a href="chap57.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>).</p>
<p>The return values of <code class="func">NF</code> (<a href="chap60.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>) and of the one argument versions of <code class="func">CF</code> (<a href="chap60.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) represent vector spaces over <span class="SimpleMath">ℚ</span>, and the return values of the two argument version of <code class="func">CF</code> (<a href="chap60.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) represent vector spaces over the field that is given as the first argument. For an abelian number field <var class="Arg">F</var> and a subfield <var class="Arg">S</var> of <var class="Arg">F</var>, a <strong class="pkg">GAP</strong> object representing <var class="Arg">F</var> as a vector space over <var class="Arg">S</var> can be constructed using <code class="func">AsField</code> (<a href="chap58.html#X7C193B7D7AFB29BE"><span class="RefLink">58.1-9</span></a>).</p>
<p>Let <var class="Arg">F</var> be the cyclotomic field <span class="SimpleMath">ℚ_n</span>, represented as a vector space over the subfield <var class="Arg">S</var>. If <var class="Arg">S</var> is the cyclotomic field <span class="SimpleMath">ℚ_m</span>, with <span class="SimpleMath">m</span> a divisor of <span class="SimpleMath">n</span>, then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns the Zumbroich basis of <var class="Arg">F</var> relative to <var class="Arg">S</var>, which consists of the roots of unity <code class="code">E(<var class="Arg">n</var>)</code>^<var class="Arg">i</var> where <var class="Arg">i</var> is an element of the list <code class="code">ZumbroichBase( <var class="Arg">n</var>, <var class="Arg">m</var> )</code> (see <code class="func">ZumbroichBase</code> (<a href="chap60.html#X7F52BEA0862E06F2"><span class="RefLink">60.3-1</span></a>)). If <var class="Arg">S</var> is an abelian number field that is not a cyclotomic field then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns a normal <var class="Arg">S</var>-basis of <var class="Arg">F</var>, i.e., a basis that is closed under the field automorphisms of <var class="Arg">F</var>.</p>
<p>Let <var class="Arg">F</var> be the abelian number field <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stab</var> )</code>, with conductor <var class="Arg">n</var>, that is itself not a cyclotomic field, represented as a vector space over the subfield <var class="Arg">S</var>. If <var class="Arg">S</var> is the cyclotomic field <span class="SimpleMath">ℚ_m</span>, with <span class="SimpleMath">m</span> a divisor of <span class="SimpleMath">n</span>, then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns the Lenstra basis of <var class="Arg">F</var> relative to <var class="Arg">S</var> that consists of the sums of roots of unity described by <code class="code">LenstraBase( <var class="Arg">n</var>, <var class="Arg">stab</var>, <var class="Arg">stab</var>, <var class="Arg">m</var> )</code> (see <code class="func">LenstraBase</code> (<a href="chap60.html#X87DB9C2C858B722A"><span class="RefLink">60.3-2</span></a>)). If <var class="Arg">S</var> is an abelian number field that is not a cyclotomic field then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns a normal <var class="Arg">S</var>-basis of <var class="Arg">F</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(8);; # a cycl. field over the rationals</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
[ 1, E(8), E(4), E(8)^3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
[ 0, 1, 0, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AsField( CF(4), CF(8) );; # a cycl. field over a cycl. field</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
[ 1, E(8) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
[ 0, 1+E(4) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AsField( Field( [ Sqrt(-2) ] ), CF(8) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># a cycl. field over a non-cycl. field</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
[ 1/2+1/2*E(8)-1/2*E(8)^2-1/2*E(8)^3,
1/2-1/2*E(8)+1/2*E(8)^2+1/2*E(8)^3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
[ E(8)+E(8)^3, E(8)+E(8)^3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= Field( [ Sqrt(-2) ] ); # a non-cycl. field over the rationals</span>
NF(8,[ 1, 3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
[ 1, E(8)+E(8)^3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
[ 0, 1 ]
</pre></div>
<p><a id="X7F52BEA0862E06F2" name="X7F52BEA0862E06F2"></a></p>
<h5>60.3-1 ZumbroichBase</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZumbroichBase</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">n</var> and <var class="Arg">m</var> be positive integers, such that <var class="Arg">m</var> divides <var class="Arg">n</var>. <code class="func">ZumbroichBase</code> returns the set of exponents <span class="SimpleMath">i</span> for which <code class="code">E(<var class="Arg">n</var>)^</code><span class="SimpleMath">i</span> belongs to the (generalized) Zumbroich basis of the cyclotomic field <span class="SimpleMath">ℚ_n</span>, viewed as a vector space over <span class="SimpleMath">ℚ_m</span>.</p>
<p>This basis is defined as follows. Let <span class="SimpleMath">P</span> denote the set of prime divisors of <var class="Arg">n</var>, <span class="SimpleMath"><var class="Arg">n</var> = ∏_{p ∈ P} p^{ν_p}</span>, and <span class="SimpleMath"><var class="Arg">m</var> = ∏_{p ∈ P} p^{μ_p}</span> with <span class="SimpleMath">μ_p ≤ ν_p</span>. Let <span class="SimpleMath">e_l =</span> <code class="code">E</code><span class="SimpleMath">(l)</span> for any positive integer <span class="SimpleMath">l</span>, and <span class="SimpleMath">{ e_{n_1}^j }_{j ∈ J} ⊗ { e_{n_2}^k }_{k ∈ K} = { e_{n_1}^j ⋅ e_{n_2}^k }_{j ∈ J, k ∈ K}</span>.</p>
<p>Then the basis is</p>
<p class="pcenter">B_{n,m} = ⨂_{p ∈ P} ⨂_{k = μ_p}^{ν_p-1} { e_{p^{k+1}}^j }_{j ∈ J_{k,p}}</p>
<p>where <span class="SimpleMath">J_{k,p} =</span></p>
<div class="pcenter"><table class="GAPDocTablenoborder">
<tr>
<td class="tdleft"><span class="SimpleMath">{ 0 }</span></td>
<td class="tdcenter">;</td>
<td class="tdleft"><span class="SimpleMath">k = 0, p = 2</span></td>
</tr>
<tr>
<td class="tdleft"><span class="SimpleMath">{ 0, 1 }</span></td>
<td class="tdcenter">;</td>
<td class="tdleft"><span class="SimpleMath">k > 0, p = 2</span></td>
</tr>
<tr>
<td class="tdleft"><span class="SimpleMath">{ 1, ..., p-1 }</span></td>
<td class="tdcenter">;</td>
<td class="tdleft"><span class="SimpleMath">k = 0, p ≠ 2</span></td>
</tr>
<tr>
<td class="tdleft"><span class="SimpleMath">{ -(p-1)/2, ..., (p-1)/2 }</span></td>
<td class="tdcenter">;</td>
<td class="tdleft"><span class="SimpleMath">k > 0, p ≠ 2</span></td>
</tr>
</table><br /><p> </p><br />
</div>
<p><span class="SimpleMath">B_{n,1}</span> is equal to the basis of <span class="SimpleMath">ℚ_n</span> over the rationals which is introduced in <a href="chapBib.html#biBZum89">[Zum89]</a>. Also the conversion of arbitrary sums of roots of unity into its basis representation, and the reduction to the minimal cyclotomic field are described in this thesis. (Note that the notation here is slightly different from that there.)</p>
<p><span class="SimpleMath">B_{n,m}</span> consists of roots of unity, it is an integral basis (that is, exactly the integral elements in <span class="SimpleMath">ℚ_n</span> have integral coefficients w.r.t. <span class="SimpleMath">B_{n,m}</span>, cf. <code class="func">IsIntegralCyclotomic</code> (<a href="chap18.html#X869750DA81EA0E67"><span class="RefLink">18.1-4</span></a>)), it is a normal basis for squarefree <span class="SimpleMath">n</span> and closed under complex conjugation for odd <span class="SimpleMath">n</span>.</p>
<p><em>Note:</em> For <span class="SimpleMath"><var class="Arg">n</var> ≡ 2 mod 4</span>, we have <code class="code">ZumbroichBase(<var class="Arg">n</var>, 1) = 2 * ZumbroichBase(<var class="Arg">n</var>/2, 1)</code> and <code class="code">List( ZumbroichBase(<var class="Arg">n</var>, 1), x -> E(<var class="Arg">n</var>)^x ) = List( ZumbroichBase(<var class="Arg">n</var>/2, 1), x -> E(<var class="Arg">n</var>/2)^x )</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ZumbroichBase( 15, 1 ); ZumbroichBase( 12, 3 );</span>
[ 1, 2, 4, 7, 8, 11, 13, 14 ]
[ 0, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ZumbroichBase( 10, 2 ); ZumbroichBase( 32, 4 );</span>
[ 2, 4, 6, 8 ]
[ 0, 1, 2, 3, 4, 5, 6, 7 ]
</pre></div>
<p><a id="X87DB9C2C858B722A" name="X87DB9C2C858B722A"></a></p>
<h5>60.3-2 LenstraBase</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LenstraBase</code>( <var class="Arg">n</var>, <var class="Arg">stabilizer</var>, <var class="Arg">super</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">n</var> and <var class="Arg">m</var> be positive integers such that <var class="Arg">m</var> divides <var class="Arg">n</var>, <var class="Arg">stabilizer</var> be a list of prime residues modulo <var class="Arg">n</var>, which describes a subfield of the <var class="Arg">n</var>-th cyclotomic field (see <code class="func">GaloisStabilizer</code> (<a href="chap60.html#X87E7313D8070B9CC"><span class="RefLink">60.2-5</span></a>)), and <var class="Arg">super</var> be a list representing a supergroup of the group given by <var class="Arg">stabilizer</var>.</p>
<p><code class="func">LenstraBase</code> returns a list <span class="SimpleMath">[ b_1, b_2, ..., b_k ]</span> of lists, each <span class="SimpleMath">b_i</span> consisting of integers such that the elements <span class="SimpleMath">∑_{j ∈ b_i}</span><code class="code">E(n)</code><span class="SimpleMath">^j</span> form a basis of the abelian number field <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stabilizer</var> )</code>, as a vector space over the <var class="Arg">m</var>-th cyclotomic field (see <code class="func">AbelianNumberField</code> (<a href="chap60.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>)).</p>
<p>This basis is an integral basis, that is, exactly the integral elements in <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stabilizer</var> )</code> have integral coefficients. (For details about this basis, see <a href="chapBib.html#biBBre97">[Bre97]</a>.)</p>
<p>If possible then the result is chosen such that the group described by <var class="Arg">super</var> acts on it, consistently with the action of <var class="Arg">stabilizer</var>, i.e., each orbit of <var class="Arg">super</var> is a union of orbits of <var class="Arg">stabilizer</var>. (A usual case is <var class="Arg">super</var><code class="code"> = </code><var class="Arg">stabilizer</var>, so there is no additional condition.</p>
<p><em>Note:</em> The <span class="SimpleMath">b_i</span> are in general not sets, since for <code class="code"><var class="Arg">stabilizer</var> = <var class="Arg">super</var></code>, the first entry is always an element of <code class="code">ZumbroichBase( <var class="Arg">n</var>, <var class="Arg">m</var> )</code>; this property is used by <code class="func">NF</code> (<a href="chap60.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>) and <code class="func">Coefficients</code> (<a href="chap61.html#X80B32F667BF6AFD8"><span class="RefLink">61.6-3</span></a>) (see <a href="chap60.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p>
<p><var class="Arg">stabilizer</var> must not contain the stabilizer of a proper cyclotomic subfield of the <var class="Arg">n</var>-th cyclotomic field, i.e., the result must describe a basis for a field with conductor <var class="Arg">n</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 24, [ 1, 19 ], [ 1, 19 ], 1 );</span>
[ [ 1, 19 ], [ 8 ], [ 11, 17 ], [ 16 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 24, [ 1, 19 ], [ 1, 5, 19, 23 ], 1 );</span>
[ [ 1, 19 ], [ 5, 23 ], [ 8 ], [ 16 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 15, [ 1, 4 ], PrimeResidues( 15 ), 1 );</span>
[ [ 1, 4 ], [ 2, 8 ], [ 7, 13 ], [ 11, 14 ] ]
</pre></div>
<p>The first two results describe two bases of the field <span class="SimpleMath">ℚ_3(sqrt{6})</span>, the third result describes a normal basis of <span class="SimpleMath">ℚ_3(sqrt{5})</span>.</p>
<p><a id="X7E4AB4B17C7BA10C" name="X7E4AB4B17C7BA10C"></a></p>
<h4>60.4 <span class="Heading">Galois Groups of Abelian Number Fields</span></h4>
<p>The field automorphisms of the cyclotomic field <span class="SimpleMath">ℚ_n</span> (see Chapter <a href="chap18.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>) are given by the linear maps <span class="SimpleMath">*k</span> on <span class="SimpleMath">ℚ_n</span> that are defined by <code class="code">E</code><span class="SimpleMath">(n)^{*k} =</span><code class="code">E</code><span class="SimpleMath">(n)^k</span>, where <span class="SimpleMath">1 ≤ k < n</span> and <code class="code">Gcd</code><span class="SimpleMath">( n, k ) = 1</span> hold (see <code class="func">GaloisCyc</code> (<a href="chap18.html#X79EE9097783128C4"><span class="RefLink">18.5-1</span></a>)). Note that this action is <em>not</em> equal to exponentiation of cyclotomics, i.e., for general cyclotomics <span class="SimpleMath">z</span>, <span class="SimpleMath">z^{*k}</span> is different from <span class="SimpleMath">z^k</span>.</p>
<p>(In <strong class="pkg">GAP</strong>, the image of a cyclotomic <span class="SimpleMath">z</span> under <span class="SimpleMath">*k</span> can be computed as <code class="code">GaloisCyc( </code><span class="SimpleMath">z, k</span><code class="code"> )</code>.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">( E(5) + E(5)^4 )^2; GaloisCyc( E(5) + E(5)^4, 2 );</span>
-2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
E(5)^2+E(5)^3
</pre></div>
<p>For <code class="code">Gcd</code><span class="SimpleMath">( n, k ) ≠ 1</span>, the map <code class="code">E</code><span class="SimpleMath">(n) ↦</span> <code class="code">E</code><span class="SimpleMath">(n)^k</span> does <em>not</em> define a field automorphism of <span class="SimpleMath">ℚ_n</span> but only a <span class="SimpleMath">ℚ</span>-linear map.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 );</span>
2
-6
</pre></div>
<p><a id="X7B55A90582E818F3" name="X7B55A90582E818F3"></a></p>
<h5>60.4-1 GaloisGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GaloisGroup</code>( <var class="Arg">F</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>The Galois group <span class="SimpleMath">Gal( ℚ_n, ℚ )</span> of the field extension <span class="SimpleMath">ℚ_n / ℚ</span> is isomorphic to the group <span class="SimpleMath">(ℤ / n ℤ)^*</span> of prime residues modulo <span class="SimpleMath">n</span>, via the isomorphism <span class="SimpleMath">(ℤ / n ℤ)^* → Gal( ℚ_n, ℚ )</span> that is defined by <span class="SimpleMath">k + n ℤ ↦ ( z ↦ z^*k )</span>.</p>
<p>The Galois group of the field extension <span class="SimpleMath">ℚ_n / L</span> with any abelian number field <span class="SimpleMath">L ⊆ ℚ_n</span> is simply the factor group of <span class="SimpleMath">Gal( ℚ_n, ℚ )</span> modulo the stabilizer of <span class="SimpleMath">L</span>, and the Galois group of <span class="SimpleMath">L / L'</span>, with <span class="SimpleMath">L'</span> an abelian number field contained in <span class="SimpleMath">L</span>, is the subgroup in this group that stabilizes <span class="SimpleMath">L'</span>. These groups are easily described in terms of <span class="SimpleMath">(ℤ / n ℤ)^*</span>. Generators of <span class="SimpleMath">(ℤ / n ℤ)^*</span> can be computed using <code class="func">GeneratorsPrimeResidues</code> (<a href="chap15.html#X7D191CF67E5018BE"><span class="RefLink">15.2-4</span></a>).</p>
<p>In <strong class="pkg">GAP</strong>, a field extension <span class="SimpleMath">L / L'</span> is given by the field object <span class="SimpleMath">L</span> with <code class="func">LeftActingDomain</code> (<a href="chap57.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>) value <span class="SimpleMath">L'</span> (see <a href="chap60.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(15);</span>
CF(15)
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GaloisGroup( f );</span>
<group with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( g ); IsCyclic( g ); IsAbelian( g );</span>
8
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Action( g, NormalBase( f ), OnPoints );</span>
Group([ (1,6)(2,4)(3,8)(5,7), (1,4,3,7)(2,8,5,6) ])
</pre></div>
<p>The following example shows Galois groups of a cyclotomic field and of a proper subfield that is not a cyclotomic field.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens1:= GeneratorsOfGroup( GaloisGroup( CF(5) ) );</span>
[ ANFAutomorphism( CF(5), 2 ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gens2:= GeneratorsOfGroup( GaloisGroup( Field( Sqrt(5) ) ) );</span>
[ ANFAutomorphism( NF(5,[ 1, 4 ]), 2 ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Order( gens1[1] ); Order( gens2[1] );</span>
4
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Sqrt(5)^gens1[1] = Sqrt(5)^gens2[1];</span>
true
</pre></div>
<p>The following example shows the Galois group of a cyclotomic field over a non-cyclotomic field.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GaloisGroup( AsField( Field( [ Sqrt(5) ] ), CF(5) ) );</span>
<group with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( g );</span>
[ ANFAutomorphism( AsField( NF(5,[ 1, 4 ]), CF(5) ), 4 ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">x:= last[1];; x^2;</span>
IdentityMapping( AsField( NF(5,[ 1, 4 ]), CF(5) ) )
</pre></div>
<p><a id="X8643D4B47A827D9D" name="X8643D4B47A827D9D"></a></p>
<h5>60.4-2 ANFAutomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ANFAutomorphism</code>( <var class="Arg">F</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">F</var> be an abelian number field and <var class="Arg">k</var> be an integer that is coprime to the conductor (see <code class="func">Conductor</code> (<a href="chap18.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>)) of <var class="Arg">F</var>. Then <code class="func">ANFAutomorphism</code> returns the automorphism of <var class="Arg">F</var> that is defined as the linear extension of the map that raises each root of unity in <var class="Arg">F</var> to its <var class="Arg">k</var>-th power.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(25);</span>
CF(25)
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha:= ANFAutomorphism( f, 2 );</span>
ANFAutomorphism( CF(25), 2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha^2;</span>
ANFAutomorphism( CF(25), 4 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Order( alpha );</span>
20
<span class="GAPprompt">gap></span> <span class="GAPinput">E(5)^alpha;</span>
E(5)^2
</pre></div>
<p><a id="X85E9E90D7FE877CC" name="X85E9E90D7FE877CC"></a></p>
<h4>60.5 <span class="Heading">Gaussians</span></h4>
<p><a id="X80BD5EAB879F096E" name="X80BD5EAB879F096E"></a></p>
<h5>60.5-1 GaussianIntegers</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GaussianIntegers</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p><code class="func">GaussianIntegers</code> is the ring <span class="SimpleMath">ℤ[sqrt{-1}]</span> of Gaussian integers. This is a subring of the cyclotomic field <code class="func">GaussianRationals</code> (<a href="chap60.html#X82F53C65802FF551"><span class="RefLink">60.1-3</span></a>).</p>
<p><a id="X7BFD33D27BFB7C5A" name="X7BFD33D27BFB7C5A"></a></p>
<h5>60.5-2 IsGaussianIntegers</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGaussianIntegers</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>is the defining category for the domain <code class="func">GaussianIntegers</code> (<a href="chap60.html#X80BD5EAB879F096E"><span class="RefLink">60.5-1</span></a>).</p>
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