%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A algext.tex GAP documentation Alexander Hulpke %% %A @(#)$Id: algext.tex,v 3.1 1994/08/31 12:14:02 mschoene Rel $ %% %Y Copyright (C) 1994, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %% This file describes algebraic extensions. %% %H $Log: algext.tex,v $ %H Revision 3.1 1994/08/31 12:14:02 mschoene %H initial revision under RCS %H %% \Chapter{Algebraic extensions of fields} \index{Field Extensions} If we adjoin a root $\alpha$ of an irreducible polynomial $p \in K[x]$ to the field $K$ we get an *algebraic extension* $K(\alpha)$, which is again a field. By Kronecker\'s construction, we may identify $K(\alpha)$ with the factor ring $K[x]/(p)$, an identification that also provides a method for computing in these extension fields. Currently {\sf GAP} only allows extension fields of fields $K$, when $K$ itself is not an extension field. As it is planned to modify the representation of field extensions to unify vector space structures and to speed up computations, {\bf All information in this chapter is subject to change in future versions}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{AlgebraicExtension} 'AlgebraicExtension( )' constructs the algebraic extension corresponding to the polynomial . must be an irreducible polynomial defined over a ``defining\'\' field . The elements of are embedded into in the canonical way. As is a field, all field functions are applicable to . Similarly, all field element functions apply to the elements of . is considered implicitely to be a field over the subfield . This means, that functions like 'Trace' and 'Norm' relative to subfields are not supported. | gap> x:=X(Rationals);;x.name:="x";; gap> p:=x^4+3*x^2+1; x^4 + 3*x^2 + 1 gap> e:=AlgebraicExtension(p); AlgebraicExtension(Rationals,x^4 + 3*x^2 + 1) gap> e.name:="e";; gap> IsField(e); true gap> y:=X(GF(2));;y.name:="y";; gap> q:=y^2+y+1; Z(2)^0*(y^2 + y + 1) gap> f:=AlgebraicExtension(q); AlgebraicExtension(GF(2),Z(2)^0*(y^2 + y + 1))| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsAlgebraicExtension} \index{test!for algebraic extension} 'IsAlgebraicExtension( )' 'IsAlgebraicExtension' returns 'true' if the object is an algebraic field extension and 'false' otherwise. More precisely, 'IsAlgebraicExtension' tests whether is an algebraic field extension record (see "Algebraic Extension Records"). So, for example, a matrix ring may in fact be a field extension, yet 'IsAlgebraicExtension' would return 'false'. | gap> IsAlgebraicExtension(e); true gap> IsAlgebraicExtension(Rationals); false| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{RootOf} \index{primitive element} 'RootOf( )' returns a root of the irreducible polynomial as element of the corresponding extension field 'AlgebraicExtension()'. This root is called the *primitive element* of this extension. % No assumptions on *which* root (as a complex number) is selected can % be made. | gap> r:=RootOf(p); RootOf(x^4 + 3*x^2 + 1) gap> r.name:="alpha";;| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Algebraic Extension Elements} \index{type!algebraic elements} \index{Operations for algebraic elements} According to Kronecker\'s construction, the elements of an algebraic extension are considered to be polynomials in the primitive element. Unless they are already in the defining field (in which case they are represented as elements of this field), they are represented by records in {\sf GAP} (see "Extension Element Records"). These records contain a representation a polynomial in the primitive element. The extension corresponding to this primitive element is the default field for the algebraic element. The usual field operations are applicable to algebraic elements. | gap> r^3/(r^2+1); -1*alpha^3-1*alpha gap> DefaultField(r^2); e| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Set functions for Algebraic Extensions} \index{membership test!for algebraic extensions} \index{in!for algebraic Elements} \index{Random!for algebraic Extensions} As algebraic extensions are fields, all set theoretic functions are applicable to algebraic elements. The following two routines are treated specially\: \vspace{5mm} 'in' tests, whether a given object is contained in an algebraic extension. The base field is embedded in the natural way into the extension. Two extensions are considered to be distinct, even if the minimal polynomial of one has a root in the other one. | gap> r in e;5 in e; true true gap> p2:=Polynomial(Rationals,MinPol(r^2)); x^2 + 3*x + 1 gap> r2:=RootOf(p2); RootOf(x^2 + 3*x + 1) gap> r2 in e; false| \vspace{5mm} 'Random' A random algebraic element is computed by taking a linear combination of the powers of the primitive element with random coefficients from the ground field. | gap> ran:=Random(e); -1*alpha^3-4*alpha^2| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsNormalExtension} \index{test!for normal extension} 'IsNormalExtension()' %'IsNormalExtension(,)' An algebraic extension field is called a *normal extension*, if it is a splitting field of the defining polynomial. The second version returns whether is a normal extension of . The first version returns whether is a normal extension of its definition field. | gap> IsNormalExtension(e); true gap> p2:=x^4+x+1;; gap> e2:=AlgebraicExtension(p2); AlgebraicExtension(Rationals,x^4 + x + 1) gap> IsNormalExtension(e2); false| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{MinpolFactors} 'MinpolFactors( )' returns the factorization of the defining polynomial of over . | gap> X(e).name:="X";; gap> MinpolFactors(e); [ X + (-1*alpha), X + (-1*alpha^3-3*alpha), X + (alpha), X + (alpha^3+3*alpha) ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{GaloisGroup for Extension Fields} \index{Galois group!of an extension field} \index{automorphism group!of an extension field} 'GaloisGroup( )' returns the Galois group of the field if is a normal extension and issues an error if not. The Galois group is a group of extension automorphisms (see "ExtensionAutomorphism"). The computation of a Galois group is computationally relatively hard, and can take significant time. | gap> g:=GaloisGroup(f); Group( ExtensionAutomorphism(AlgebraicExtension(GF(2),Z(2)^0*(y^ 2 + y + 1)),RootOf(Z(2)^0*(y^2 + y + 1))+Z(2)^0) ) gap> h:=GaloisGroup(e); Group( ExtensionAutomorphism(e,alpha^3+ 3*alpha), ExtensionAutomorphism(e, -1*alpha), ExtensionAutomorphism(e,-1*alpha^3-3*alpha) ) gap> Size(h); 4 gap> AbelianInvariants(h); [ 2, 2 ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ExtensionAutomorphism} \index{field homomorphisms!of algebraic extensions} 'ExtensionAutomorphism( , )' is the automorphism of the extension , that maps the primitive root of to . As it is a field automorphism, section "Field Homomorphisms" applies. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Field functions for Algebraic Extensions} \index{Norm!for algebraic extensions} \index{Trace!for algebraic extensions} \index{DefaultField!for algebraic extensions} \index{Field!for algebraic extensions} As already mentioned, algebraic extensions are fields. Thus all field functions like 'Norm' and 'Trace' are applicable. | gap> Trace(r^4+2*r); 14 gap> Norm(ran); 305| 'DefaultField' always returns the algebraic extension, which contains the primitive element by which the number is represented, see "Algebraic Extension Elements". | gap> DefaultField(r^2); e| As subfields are not yet supported, 'Field' will issue an error, if several elements are given, or if the element is not a primitive element for its default field. You can create a polynomial ring over an algebraic extension to which all functions described in "Ring Functions for Polynomial Rings" can be applied, for example you can factor polynomials. Factorization is done --- depending on the polynomial --- by factoring the squarefree norem or using a hensel lift (with possibly added lattice reduction) as described in \cite{Abb89}, using bounds from \cite{BTW93}. | gap> X(e).name:="X";; gap> p2:=EmbeddedPolynomial(PolynomialRing(e),p2); X^2 + 3*X + 1 gap> Factors(p2); [ X + (-1*alpha^2), X + (alpha^2+3) ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Algebraic Extension Records} \index{record fields!for algebraic extension fields} Since every algebraic extension is a field, it is represented as a record. This record contains all components, a field record will contain (see "Field Records"). Additionally, it contains the components 'isAlgebraicExtension', 'minpol', 'primitiveElm' and may contain the components 'isNormalExtension', 'minpolFactors' and 'galoisType'. 'isAlgebraicExtension': \\ is always 'true'. This indicates that is an algebraic extension. 'minpol': \\ is the defining polynomial of . 'primitiveElm': \\ contains 'RootOf(.minpol)'. 'isNormalExtension': \\ indicates, whether is a normal extension field. 'minpolFactors': \\ contains a factorization of '.minpol' over . 'galoisType': \\ contains the Galois type of the normal closure of . See section "GaloisType". %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Extension Element Records} \index{record fields!for extension elements} Elements of an algebraic extension are represented by a record. The record for the element of contains the components 'isAlgebraicElement', 'domain' and 'coefficients'\: 'isAlgebraicElement': \\ is always 'true', and indicates, that is an algebraic element. 'domain': \\ contains . 'coefficients': \\ contains the coefficients of as a polynomial in the primitive root of . %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsAlgebraicElement} \index{test!for algebraic element} 'IsAlgebraicElement( )' returns 'true' if obj is an algebraic element, i.e., an element of an algebraic extension, that is not in the defining field, and 'false' otherwise. | gap> IsAlgebraicElement(r); true gap> IsAlgebraicElement(3); false| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Algebraic extensions of the Rationals} The following sections describe functions that are specific to algebraic extensions of ${Q\mskip-11mu\prime\,\,}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{DefectApproximation} \index{defect} 'DefectApproximation( )' computes a multiple of the defect of the basis of , given by the powers of the primitive element. The *defect* indicates, which denominator is necessary in the coefficients, to express algebraic integers in as a linear combination of the base of . 'DefectApproximation' takes the maximal square in the discriminant as a first approximation, and then uses Berwicks and Hesses method (see \cite{Bra89}) to improve this approximation. The number returned is not neccessarily the defect, but may be a proper multiple of it. | gap> DefectApproximation(e); 1| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{GaloisType} \index{Galois} 'GaloisType( )' \\ 'Galois( )' The first version returns the number of the permutation isomorphism type of the Galois group of the normal closure of , considered as a transitive permutation group of the roots of the defining polynomial (see "The Transitive Groups Library"). The second version returns the Galois type of the splitting field of . Identification is done by factoring appropriate Galois resolvents as proposed in \cite{MS85}. This function is provided for rational polynomials of degree up to 15. However, it may be not feasible to call this function for polynomials of degree 14 or 15, as the involved computations may be enormous. For some polynomials of degree 14, a complete discrimination is not yet possible, as it would require computations, that are not feasible with current factoring methods. | gap> GaloisType(e); 2 gap> TransitiveGroup(e.degree,2); 2[x]2 = E(4)| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ProbabilityShapes} 'ProbabilityShapes( )' returns a list of numbers, which contains most likely the isomorphism type of the galois group of (see "GaloisType"). This routine only applies the cycle structure test according to Tschebotareff\'s theorem. Accordingly, it is very fast, but the result is not guaranteed to be correct. | gap> ProbabilityShapes(e.minpol); [ 2 ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{DecomPoly} \index{decomposition!of polynomials} \index{ideal decomposition} 'DecomPoly( )'\\ 'DecomPoly( , \"all\" )' returns an ideal decomposition of the polynomial . An ideal decomposition is given by two polynomials and , such that $pol$ divides $(g\circ h)$. By the Galois correspondence any ideal decomposition corresponds to a block system of the Galois group. The polynomial defines a subfield $K(\beta)$ of $K(\alpha)$ with $h(\alpha)=\beta$. The first form finds one ideal decomposition, while the second form finds all possible different ideal decompositions (i.e. all subfields). | gap> d:=DecomPoly(e.minpol); [ ] gap> p:=x^10-10*x^9+20*x^8+25*x^7+(-1885/8)*x^6+(109/4)*x^5 > +(3635/8)*x^4+(4535/16)*x^3+(-456615/256)*x^2+(135315/128)*x > +(-64681/128);; gap> d:=DecomPoly(p,"all"); [ [ x^5 - 608253837790304032960320*x^4 + 102513277599629696329909384806667776261204935040*x^3 - 549049420457453617648141380902484571131598386028760933387027\ 6218920960*x^2 + 286387493717897513685666250304380673273734943195175651689786\ 74636841864755780846410697052160*x - 173134960201194027097404571687818402186256514770030882717264\ 9557460578764008149662018624414041261233616423188430848, 8772892716132663808*x^9 - 52775093681915390720*x^8 - 205982735485386544256*x^7 + 1328709758167472769984*x^6 - 2621982207697769174432*x^5 - 7339219574720832743376*x^4 + 9552106782081018334600*x^3 + 15478181913028444442212*x^2 - 46541684909437606849432*x + 16156582976797641690520 ] ] gap> Degree(Value(d[1][1],d[1][2])/f); 35|