<html><head><title>[ref] 56 Fields and Division Rings</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href="../index.htm">Top</a>] [<a href = "chapters.htm">Up</a>] [<a href ="CHAP055.htm">Previous</a>] [<a href ="CHAP057.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>56 Fields and Division Rings</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP056.htm#SECT001">Generating Fields</a> <li> <A HREF="CHAP056.htm#SECT002">Subfields of Fields</a> <li> <A HREF="CHAP056.htm#SECT003">Galois Action</a> </ol><p> <p> <a name = "I0"></a> <a name = "I1"></a> A <strong>division ring</strong> is a ring (see Chapter <a href="CHAP054.htm">Rings</a>) in which every non-zero element has an inverse. The most important class of division rings are the commutative ones, which are called <strong>fields</strong>. <p> <font face="Gill Sans,Helvetica,Arial">GAP</font> supports finite fields (see Chapter <a href="CHAP057.htm">Finite Fields</a>) and abelian number fields (see Chapter <a href="CHAP058.htm">Abelian Number Fields</a>), in particular the field of rationals (see Chapter <a href="CHAP016.htm">Rational Numbers</a>). <p> This chapter describes the general <font face="Gill Sans,Helvetica,Arial">GAP</font> functions for fields and division rings. <p> If a field <var>F</var> is a subfield of a commutative ring <var>C</var>, <var>C</var> can be considered as a vector space over the (left) acting domain <var>F</var> (see Chapter <a href="CHAP059.htm">Vector Spaces</a>). In this situation, we call <var>F</var> the <strong>field of definition</strong> of <var>C</var>. <p> Each field in <font face="Gill Sans,Helvetica,Arial">GAP</font> is represented as a vector space over a subfield (see <a href="CHAP056.htm#SSEC001.2">IsField</a>), thus each field is in fact a field extension in a natural way, which is used by functions such as <code>Norm</code> and <code>Trace</code> (see <a href="CHAP056.htm#SECT003">Galois Action</a>). <p> <p> <h2><a name="SECT001">56.1 Generating Fields</a></h2> <p><p> <a name = "SSEC001.1"></a> <li><code>IsDivisionRing( </code><var>D</var><code> ) C</code> <p> A <strong>division ring</strong> in <font face="Gill Sans,Helvetica,Arial">GAP</font> is a nontrivial associative algebra <var>D</var> with a multiplicative inverse for each nonzero element. In <font face="Gill Sans,Helvetica,Arial">GAP</font> every division ring is a vector space over a division ring (possibly over itself). Note that being a division ring is thus not a property that a ring can get, because a ring is usually not represented as a vector space. <p> The field of coefficients is stored as <code>LeftActingDomain( </code><var>D</var><code> )</code>. <p> <a name = "SSEC001.2"></a> <li><code>IsField( </code><var>D</var><code> ) P</code> <p> A <strong>field</strong> is a commutative division ring (see <a href="CHAP056.htm#SSEC001.1">IsDivisionRing</a> and <a href="CHAP033.htm#SSEC004.9">IsCommutative</a>). <p> <pre> gap> IsField( GaloisField(16) ); # the field with 16 elements true gap> IsField( Rationals ); # the field of rationals true gap> q:= QuaternionAlgebra( Rationals );; # a noncommutative division ring gap> IsField( q ); IsDivisionRing( q ); false true gap> mat:= [ [ 1 ] ];; a:= Algebra( Rationals, [ mat ] );; gap> IsDivisionRing( a ); # an algebra not constructed as a division ring false </pre> <p> <a name = "SSEC001.3"></a> <li><code>Field( </code><var>z</var><code>, ... ) F</code> <li><code>Field( </code><var>list</var><code> ) F</code> <li><code>Field( </code><var>F</var><code>, </code><var>list</var><code> ) F</code> <p> <code>Field</code> returns the smallest field <i>K</i> that contains all the elements <i>z</i> , …, or the smallest field <i>K</i> that contains all elements in the list <var>list</var>. If no subfield <var>F</var> is given, <i>K</i> is constructed as a field over itself, i.e. the left acting domain of <i>K</i> is <i>K</i>. In the third form, <code>Field</code> constructs the field generated by the field <var>F</var> and the elements in the list <var>list</var>, as a vector space over <var>F</var>. <p> <a name = "SSEC001.4"></a> <li><code>DefaultField( </code><var>z</var><code>, ... ) F</code> <li><code>DefaultField( </code><var>list</var><code> ) F</code> <p> <code>DefaultField</code> returns a field <i>K</i> that contains all the elements <i>z</i> , …, or a field <i>K</i> that contains all elements in the list <var>list</var>. <p> This field need not be the smallest field in which the elements lie, cf. <code>Field</code> (see <a href="CHAP056.htm#SSEC001.3">Field</a>). For example, for elements from cyclotomic fields <code>DefaultField</code> returns the smallest cyclotomic field in which the elements lie, but the elements may lie in a smaller number field which is not a cyclotomic field. <p> <pre> gap> Field( Z(4) ); Field( [ Z(4), Z(8) ] ); # finite fields GF(2^2) GF(2^6) gap> Field( E(9) ); Field( CF(4), [ E(9) ] ); # abelian number fields CF(9) AsField( GaussianRationals, CF(36) ) gap> f1:= Field( EB(5) ); f2:= DefaultField( EB(5) ); NF(5,[ 1, 4 ]) CF(5) gap> f1 = f2; IsSubset( f2, f1 ); false true </pre> <p> <a name = "SSEC001.5"></a> <li><code>DefaultFieldByGenerators( [ </code><var>z</var><code>, ... ] ) O</code> <p> returns the default field containing the elements <var>z</var>,…. This field may be bigger than the smallest field containing these elements. <p> <a name = "SSEC001.6"></a> <li><code>GeneratorsOfDivisionRing( </code><var>D</var><code> ) A</code> <p> generators with respect to addition, multiplication, and taking inverses (the identity cannot be omitted ...) <p> <a name = "SSEC001.7"></a> <li><code>GeneratorsOfField( </code><var>F</var><code> ) A</code> <p> generators with respect to addition, multiplication, and taking inverses. This attribute is the same as <code>GeneratorsOfDivisionRing</code> (see <a href="CHAP056.htm#SSEC001.6">GeneratorsOfDivisionRing</a>). <p> <a name = "SSEC001.8"></a> <li><code>DivisionRingByGenerators( [ </code><var>z</var><code>, ... ] ) O</code> <li><code>DivisionRingByGenerators( </code><var>F</var><code>, [ </code><var>z</var><code>, ... ] ) O</code> <p> The first version returns a division ring as vector space over <code>FieldOverItselfByGenerators( </code><var>gens</var><code> )</code>. <p> <a name = "SSEC001.9"></a> <li><code>AsDivisionRing( </code><var>C</var><code> ) O</code> <li><code>AsDivisionRing( </code><var>F</var><code>, </code><var>C</var><code> ) O</code> <a name = "SSEC001.9"></a> <li><code>AsField( </code><var>C</var><code> ) O</code> <li><code>AsField( </code><var>F</var><code>, </code><var>C</var><code> ) O</code> <p> If the collection <var>C</var> can be regarded as a division ring then <code>AsDivisionRing( </code><var>C</var><code> )</code> is the division ring that consists of the elements of <var>C</var>, viewed as a vector space over its prime field; otherwise <code>fail</code> is returned. <p> In the second form, if <var>F</var> is a division ring contained in <var>C</var> then the returned division ring is viewed as a vector space over <var>F</var>. <p> <code>AsField</code> is just a synonym for <code>AsDivisionRing</code>. <p> <p> <h2><a name="SECT002">56.2 Subfields of Fields</a></h2> <p><p> <a name = "SSEC002.1"></a> <li><code>Subfield( </code><var>F</var><code>, </code><var>gens</var><code> ) F</code> <a name = "SSEC002.1"></a> <li><code>SubfieldNC( </code><var>F</var><code>, </code><var>gens</var><code> ) F</code> <p> Constructs the subfield of <var>F</var> generated by <var>gens</var>. <p> <a name = "SSEC002.2"></a> <li><code>FieldOverItselfByGenerators( [ </code><var>z</var><code>, ... ] ) O</code> <p> This operation is needed for the call of <code>Field</code> or <code>FieldByGenerators</code> without explicitly given subfield, in order to construct a left acting domain for such a field. <p> <a name = "SSEC002.3"></a> <li><code>PrimitiveElement( </code><var>D</var><code> ) A</code> <p> is an element of <var>D</var> that generates <var>D</var> as a division ring together with the left acting domain. <p> <a name = "SSEC002.4"></a> <li><code>PrimeField( </code><var>D</var><code> ) A</code> <p> The <strong>prime field</strong> of a division ring <var>D</var> is the smallest field which is contained in <var>D</var>. For example, the prime field of any field in characteristic zero is isomorphic to the field of rational numbers. <p> <a name = "SSEC002.5"></a> <li><code>IsPrimeField( </code><var>D</var><code> ) P</code> <p> A division ring is a prime field if it is equal to its prime field (see <a href="CHAP056.htm#SSEC002.4">PrimeField</a>). <p> <a name = "SSEC002.6"></a> <li><code>DegreeOverPrimeField( </code><var>F</var><code> ) A</code> <p> is the degree of the field <var>F</var> over its prime field (see <a href="CHAP056.htm#SSEC002.4">PrimeField</a>). <p> <a name = "SSEC002.7"></a> <li><code>DefiningPolynomial( </code><var>F</var><code> ) A</code> <p> is the defining polynomial of the field <var>F</var> as a field extension over the left acting domain of <var>F</var>. A root of the defining polynomial can be computed with <code>RootOfDefiningPolynomial</code> (see <a href="CHAP056.htm#SSEC002.8">RootOfDefiningPolynomial</a>). <p> <a name = "SSEC002.8"></a> <li><code>RootOfDefiningPolynomial( </code><var>F</var><code> ) A</code> <p> is a root in the field <var>F</var> of its defining polynomial as a field extension over the left acting domain of <var>F</var>. The defining polynomial can be computed with <code>DefiningPolynomial</code> (see <a href="CHAP056.htm#SSEC002.7">DefiningPolynomial</a>). <p> <a name = "SSEC002.9"></a> <li><code>FieldExtension( </code><var>F</var><code>, </code><var>poly</var><code> ) O</code> <p> is the field obtained on adjoining a root of the irreducible polynomial <var>poly</var> to the field <var>F</var>. <p> <a name = "SSEC002.10"></a> <li><code>Subfields( </code><var>F</var><code> ) A</code> <p> is the set of all subfields of the field <var>F</var>. <p> <p> <h2><a name="SECT003">56.3 Galois Action</a></h2> <p><p> Let <i>L</i> > <i>K</i> be a field extension of finite degree. Then to each element α ∈ <i>L</i>, we can associate a <i>K</i>-linear mapping ϕ<sub>α</sub> on <i>L</i>, and for a fixed <i>K</i>-basis of <i>L</i>, we can associate to α the matrix <i>M</i><sub>α</sub> (over <i>K</i>) of this mapping. <p> The <strong>norm</strong> of α is defined as the determinant of <i>M</i><sub>α</sub>, the <strong>trace</strong> of α is defined as the trace of <i>M</i><sub>α</sub>, the <strong>minimal polynomial</strong> μ<sub>α</sub> and the <strong>trace polynomial</strong> χ<sub>α</sub> of α are defined as the minimal polynomial (see <a href="CHAP056.htm#SSEC003.2">MinimalPolynomial!over a field</a>) and the characteristic polynomial (see <a href="CHAP024.htm#SSEC012.1">CharacteristicPolynomial</a> and <a href="CHAP056.htm#SSEC003.3">TracePolynomial</a>) of <i>M</i><sub>α</sub>. (Note that μ<sub>α</sub> depends only on <i>K</i> whereas χ<sub>α</sub> depends on both <i>L</i> and <i>K</i>.) <p> Thus norm and trace of α are elements of <i>K</i>, and μ<sub>α</sub> and χ<sub>α</sub> are polynomials over <i>K</i>, χ<sub>α</sub> being a power of μ<sub>α</sub>, and the degree of χ<sub>α</sub> equals the degree of the field extension <i>L</i> > <i>K</i>. <p> The <strong>conjugates</strong> of α in <i>L</i> are those roots of χ<sub>α</sub> (with multiplicity) that lie in <i>L</i>; note that if only <i>L</i> is given, there is in general no way to access the roots outside <i>L</i>. <p> Analogously, the <strong>Galois group</strong> of the extension <i>L</i> > <i>K</i> is defined as the group of all those field automorphisms of <i>L</i> that fix <i>K</i> pointwise. <p> If <i>L</i> > <i>K</i> is a Galois extension then the conjugates of α are all roots of χ<sub>α</sub> (with multiplicity), the set of conjugates equals the roots of μ<sub>α</sub>, the norm of α equals the product and the trace of α equals the sum of the conjugates of α, and the Galois group in the sense of the above definition equals the usual Galois group, <p> Note that <code>MinimalPolynomial( </code><var>F</var><code>, </code><var>z</var><code> )</code> is a polynomial <strong>over</strong> <var>F</var>, whereas <code>Norm( </code><var>F</var><code>, </code><var>z</var><code> )</code> is the norm of the element <var>z</var> <strong>in</strong> <var>F</var> w.r.t. the field extension <i>F</i> > <tt>LeftActingDomain</tt><tt>(</tt> <i>F</i> <tt>)</tt>. <p> <a name = "I2"></a> <a name = "SSEC003.1"></a> <li><code>GaloisGroup( </code><var>F</var><code> ) A</code> <p> The <strong>Galois group</strong> of a field <var>F</var> is the group of all field automorphisms of <var>F</var> that fix the subfield <i>K</i> = <tt>LeftActingDomain</tt><tt>(</tt> <i>F</i> <tt>)</tt> pointwise. <p> Note that the field extension <i>F</i> > <i>K</i> need <strong>not</strong> be a Galois extension. <p> <pre> gap> g:= GaloisGroup( AsField( GF(2^2), GF(2^12) ) );; gap> Size( g ); IsCyclic( g ); 6 true gap> h:= GaloisGroup( CF(60) );; gap> Size( h ); IsAbelian( h ); 16 true </pre> <p> <a name = "SSEC003.2"></a> <li><code>MinimalPolynomial( </code><var>F</var><code>, </code><var>z</var><code>[, </code><var>ind</var><code>] ) O</code> <p> returns the minimal polynomial of <var>z</var> over the field <var>F</var>. This is a generator of the ideal in <i>F</i> [<i>x</i>] of all polynomials which vanish on <var>z</var>. (This definition is consistent with the general definition of <code>MinimalPolynomial</code> for rings, see <a href="CHAP064.htm#SSEC008.1">MinimalPolynomial</a>.) <p> <pre> gap> MinimalPolynomial( Rationals, E(8) ); x_1^4+1 gap> MinimalPolynomial( CF(4), E(8) ); x_1^2+(-E(4)) gap> MinimalPolynomial( CF(8), E(8) ); x_1+(-E(8)) </pre> <p> <a name = "SSEC003.3"></a> <li><code>TracePolynomial( </code><var>L</var><code>, </code><var>K</var><code>, </code><var>z</var><code>[, </code><var>inum</var><code>] ) O</code> <p> returns the polynomial that is the product of (<i>X</i> − <i>c</i>) where <i>c</i> runs over the conjugates of <var>z</var> in the field extension <var>L</var> over <var>K</var>. The polynomial is returned as a univariate polynomial over <var>K</var> in the indeterminate number <var>inum</var> (defaulting to 1). <p> This polynomial is sometimes also called the <strong>characteristic polynomial</strong> of <var>z</var> w.r.t. the field extension <i>L</i> > <i>K</i> . Therefore methods are installed for <code>CharacteristicPolynomial</code> (see <a href="CHAP024.htm#SSEC012.1">CharacteristicPolynomial</a>) that call <code>TracePolynomial</code> in the case of field extensions. <p> <a name = "I3"></a> <pre> gap> TracePolynomial( CF(8), Rationals, E(8) ); x_1^4+1 gap> TracePolynomial( CF(16), Rationals, E(8) ); x_1^8+2*x_1^4+1 </pre> <p> <a name = "SSEC003.4"></a> <li><code>Norm( </code><var>z</var><code> ) A</code> <li><code>Norm( </code><var>L</var><code>, </code><var>z</var><code> ) O</code> <li><code>Norm( </code><var>L</var><code>, </code><var>K</var><code>, </code><var>z</var><code> ) O</code> <p> <code>Norm</code> returns the norm of the field element <var>z</var>. If two fields <var>L</var> and <var>K</var> are given then the norm is computed w.r.t. the field extension <i>L</i> > <i>K</i> , if only one field <var>L</var> is given then <code>LeftActingDomain( </code><var>L</var><code> )</code> is taken as default for the subfield <var>K</var>, and if no field is given then <code>DefaultField( </code><var>z</var><code> )</code> is taken as default for <var>L</var>. <p> <a name = "SSEC003.5"></a> <li><code>Trace( </code><var>z</var><code> ) A</code> <li><code>Trace( </code><var>mat</var><code> ) A</code> <li><code>Trace( </code><var>L</var><code>, </code><var>z</var><code> ) O</code> <li><code>Trace( </code><var>L</var><code>, </code><var>K</var><code>, </code><var>z</var><code> ) O</code> <p> <code>Trace</code> returns the trace of the field element <var>z</var>. If two fields <var>L</var> and <var>K</var> are given then the trace is computed w.r.t. the field extension <i>L</i> > <i>K</i> , if only one field <var>L</var> is given then <code>LeftActingDomain( </code><var>L</var><code> )</code> is taken as default for the subfield <var>K</var>, and if no field is given then <code>DefaultField( </code><var>z</var><code> )</code> is taken as default for <var>L</var>. <p> The <strong>trace of a matrix</strong> is the sum of its diagonal entries. Note that this is <strong>not</strong> compatible with the definition of <code>Trace</code> for field elements, so the one-argument version is not suitable when matrices shall be regarded as field elements. <p> <a name = "SSEC003.6"></a> <li><code>Conjugates( </code><var>z</var><code> ) A</code> <li><code>Conjugates( </code><var>L</var><code>, </code><var>z</var><code> ) O</code> <li><code>Conjugates( </code><var>L</var><code>, </code><var>K</var><code>, </code><var>z</var><code> ) O</code> <p> <code>Conjugates</code> returns the list of <strong>conjugates</strong> of the field element <var>z</var>. If two fields <var>L</var> and <var>K</var> are given then the conjugates are computed w.r.t. the field extension <i>L</i> > <i>K</i> , if only one field <var>L</var> is given then <code>LeftActingDomain( </code><var>L</var><code> )</code> is taken as default for the subfield <var>K</var>, and if no field is given then <code>DefaultField( </code><var>z</var><code> )</code> is taken as default for <var>L</var>. <p> The result list will contain duplicates if <var>z</var> lies in a proper subfield of <var>L</var>, respectively of the default field of <var>z</var>. The result list need not be sorted. <p> <pre> gap> Norm( E(8) ); Norm( CF(8), E(8) ); 1 1 gap> Norm( CF(8), CF(4), E(8) ); -E(4) gap> Norm( AsField( CF(4), CF(8) ), E(8) ); -E(4) gap> Trace( E(8) ); Trace( CF(8), CF(8), E(8) ); 0 E(8) gap> Conjugates( CF(8), E(8) ); [ E(8), E(8)^3, -E(8), -E(8)^3 ] gap> Conjugates( CF(8), CF(4), E(8) ); [ E(8), -E(8) ] gap> Conjugates( CF(16), E(8) ); [ E(8), E(8)^3, -E(8), -E(8)^3, E(8), E(8)^3, -E(8), -E(8)^3 ] </pre> <p> The default methods for field elements are as follows. <code>MinimalPolynomial</code> solves a system of linear equations, <code>TracePolynomial</code> computes the appropriate power of the minimal polynomial, <code>Norm</code> and <code>Trace</code> values are obtained as coefficients of the characteristic polynomial, and <code>Conjugates</code> uses the factorization of the characteristic polynomial. <p> For elements in finite fields and cyclotomic fields, one wants to do the computations in a different way since the field extensions in question are Galois extensions, and the Galois groups are well-known in these cases. More general, if a field is in the category <code>IsFieldControlledByGaloisGroup</code> then the default methods are the following. <code>Conjugates</code> returns the sorted list of images (with multiplicity) of the element under the Galois group, <code>Norm</code> computes the product of the conjugates, <code>Trace</code> computes the sum of the conjugates, <code>TracePolynomial</code> and <code>MinimalPolynomial</code> compute the product of linear factors <i>x</i> − <i>c</i> with <i>c</i> ranging over the conjugates and the set of conjugates, respectively. <p> <a name = "SSEC003.7"></a> <li><code>NormalBase( </code><var>F</var><code> ) A</code> <li><code>NormalBase( </code><var>F</var><code>, </code><var>elm</var><code> ) O</code> <p> Let <var>F</var> be a field that is a Galois extension of its subfield <code>LeftActingDomain( </code><var>F</var><code> )</code>. Then <code>NormalBase</code> returns a list of elements in <var>F</var> that form a normal basis of <var>F</var>, that is, a vector space basis that is closed under the action of the Galois group (see <a href="CHAP056.htm#SSEC003.1">GaloisGroup!of field</a>) of <var>F</var>. <p> If a second argument <var>elm</var> is given, it is used as a hint for the algorithm to find a normal basis with the algorithm described in <a href="biblio.htm#Art68"><cite>Art68</cite></a>. <p> <pre> gap> NormalBase( CF(5) ); [ -E(5), -E(5)^2, -E(5)^3, -E(5)^4 ] gap> NormalBase( CF(4) ); [ 1/2-1/2*E(4), 1/2+1/2*E(4) ] gap> NormalBase( GF(3^6) ); [ Z(3^6)^2, Z(3^6)^6, Z(3^6)^18, Z(3^6)^54, Z(3^6)^162, Z(3^6)^486 ] gap> NormalBase( GF( GF(8), 2 ) ); [ Z(2^6), Z(2^6)^8 ] </pre> <p> <p> [<a href="../index.htm">Top</a>] [<a href = "chapters.htm">Up</a>] [<a href ="CHAP055.htm">Previous</a>] [<a href ="CHAP057.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <font face="Gill Sans,Helvetica,Arial">GAP 4 manual<br>December 2008 </font></body></html>
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