%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A field.tex GAP documentation Martin Schoenert %% %A @(#)$Id: field.tex,v 3.12 1994/06/03 08:57:20 mschoene Rel $ %% %Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %% This file describes the operators and functions of finite field elements. %% %H $Log: field.tex,v $ %H Revision 3.12 1994/06/03 08:57:20 mschoene %H changed a few things to avoid LaTeX warnings %H %H Revision 3.11 1994/03/18 14:22:47 ahulpke %H Fixed 'Conjugates' and its relatives for nonabelian extensions %H %H Revision 3.10 1993/02/19 10:48:42 gap %H adjustments in line length and spelling %H %H Revision 3.9 1993/02/15 09:49:57 felsch %H another example fixed %H %H Revision 3.8 1993/02/13 08:49:49 felsch %H error messages in examples fixed %H %H Revision 3.7 1993/02/12 17:19:25 felsch %H examples adjusted to line length 72 %H %H Revision 3.6 1992/05/25 18:13:25 martin %H fixed a typo in "Field Homomorphisms" %H %H Revision 3.5 1992/04/06 15:19:57 martin %H fixed some more typos %H %H Revision 3.4 1992/03/25 15:37:32 martin %H added new sections for field homomorphisms %H %H Revision 3.3 1992/03/11 15:50:48 sam %H renamed chapter "Number Fields" to "Subfields of Cyclotomic Fields" %H %H Revision 3.2 1991/12/30 12:07:53 martin %H fixed a few incorrect references %H %H Revision 3.1 1991/12/30 11:45:29 martin %H changed incorrect reference to "CyclotomicField" %H %H Revision 3.0 1991/12/27 16:10:27 martin %H initial revision under RCS %H %% \Chapter{Fields} Fields are important algebraic domains. Mathematically a *field* is a commutative ring $F$ (see chapter "Rings"), such that every element except $0$ has a multiplicative inverse. Thus $F$ has two operations '+' and '\*' called addition and multiplication. $(F,+)$ must be an abelian group, whose identity is called $0_F$. $(F-\{0_F\},\*)$ must be an abelian group, whose identity element is called $1_F$. {\GAP} supports the field of rationals (see "Rationals"), subfields of cyclotomic fields (see "Subfields of Cyclotomic Fields"), and finite fields (see "Finite Fields"). This chapter begins with sections that describe how to test whether a domain is a field (see "IsField"), how to find the smallest field and the default field in which a list of elements lies (see "Field" and "DefaultField"), and how to view a field over a subfield (see "Fields over Subfields"). The next sections describes the operation applicable to field elements (see "Comparisons of Field Elements" and "Operations for Field Elements"). The next sections describe the functions that are applicable to fields (see "GaloisGroup") and their elements (see "Conjugates", "Norm", "Trace", "CharPol", and "MinPol"). The following sections describe homomorphisms of fields (see "Field Homomorphisms", "IsFieldHomomorphism", "KernelFieldHomomorphism", "Mapping Functions for Field Homomorphisms"). The last section describes how fields are represented internally (see "Field Records"). Fields are domains, so all functions that are applicable to all domains are also applicable to fields (see chapter "Domains"). All functions for fields are in 'LIBNAME/\"field.g\"'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsField} 'IsField( )' 'IsField' returns 'true' if the object is a field and 'false' otherwise. More precisely 'IsField' tests whether is a field record (see "Field Records"). So, for example, a matrix group may in fact be a field, yet 'IsField' would return 'false'. | gap> IsField( GaloisField(16) ); true gap> IsField( CyclotomicField(9) ); true gap> IsField( rec( isDomain := true, isField := true ) ); true # it is possible to fool 'IsField' gap> IsField( AsRing( Rationals ) ); false # though this ring is, as a set, still 'Rationals' | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Field} 'Field( ,.. )' 'Field( )' In the first form 'Field' returns the smallest field that contains all the elements ,.. etc. In the second form 'Field' returns the smallest field that contains all the elements in the list . If any element is not an element of a field or the elements lie in no common field an error is raised. | gap> Field( Z(4) ); GF(2^2) gap> Field( E(9) ); CF(9) gap> Field( [ Z(4), Z(9) ] ); Error, CharFFE: must be a finite field element, vector, or matrix gap> Field( [ E(4), E(9) ] ); CF(36) | 'Field' differs from 'DefaultField' (see "DefaultField") in that it returns the smallest field in which the elements lie, while 'DefaultField' may return a larger field if that makes sense. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{DefaultField} 'DefaultField( ,.. )' 'DefaultField( )' In the first form 'DefaultField' returns the default field that contains all the elements ,.. etc. In the second form 'DefaultField' returns the default field that contains all the elements in the list . If any element is not an element of a field or the elements lie in no common field an error is raised. The field returned by 'DefaultField' need not be the smallest field in which the elements lie. For example for elements from cyclotomic fields 'DefaultField' may return the smallest cyclotomic field in which the elements lie, which need not be the smallest field overall, because the elements may in fact lie in a smaller number field which is not a cyclotomic field. For the exact definition of the default field of a certain type of elements read the chapter describing this type (see "Finite Fields" and "Subfields of Cyclotomic Fields"). 'DefaultField' is used by 'Conjugates', 'Norm', 'Trace', 'CharPol', and 'MinPol' (see "Conjugates", "Norm", "Trace", "CharPol", and "MinPol") if no explicit field is given. | gap> DefaultField( Z(4) ); GF(2^2) gap> DefaultField( E(9) ); CF(9) gap> DefaultField( [ Z(4), Z(9) ] ); Error, CharFFE: must be a finite field element, vector, or matrix gap> DefaultField( [ E(4), E(9) ] ); CF(36) | 'Field' (see "Field") differs from 'DefaultField' in that it returns the smallest field in which the elements lie, while 'DefaultField' may return a larger field if that makes sense. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Fields over Subfields} ' / ' The quotient operator '/' evaluates to a new field . This field has the same elements as , i.e., is a domain equal to . However is viewed as a field over the field , which must be a subfield of . What subfield a field is viewed over determines its Galois group. As described in "GaloisGroup" the Galois group is the group of field automorphisms that leave the subfield fixed. It also influences the results of "Norm", "Trace", "CharPol", and "MinPol", because they are defined in terms of the Galois group. | gap> F := GF(2^12); GF(2^12) gap> G := GF(2^2); GF(2^2) gap> Q := F / G; GF(2^12)/GF(2^2) gap> Norm( F, Z(2^6) ); Z(2)^0 gap> Norm( Q, Z(2^6) ); Z(2^2)^2 | The operator '/' calls '.operations./( , )'. The default function called this way is 'FieldOps./', which simply makes a copy of and enters into the record component '.field' (see "Field Records"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Comparisons of Field Elements} ' = ' \\ ' \<> ' The equality operator '=' evaluates to 'true' if the two field elements and are equal, and to 'false' otherwise. The inequality operator '\<>' evaluates to 'true' if the two field elements and are not equal, and to 'false' otherwise. Note that any two field elements can be compared, even if they do not lie in compatible fields. In this case they cn, of course, never be equal. For each type of fields the equality of those field elements is given in the respective chapter. Note that you can compare field elements with elements of other types; of course they are never equal. ' \<\ ' \\ ' \<= ' \\ ' > ' \\ ' >= ' The operators '\<', '\<=', '>', and '>=' evaluate to 'true' if the field element is less than, less than or equal to, greater than, or greater than or equal to the field element . For each type of fields the definition of the ordering of those field elements is given in the respective chapter. The ordering of field elements is as follows. Rationals are smallest, next are cyclotomics, followed by finite field elements. Note that you can compare field elements with elements of other types; they are smaller than everything else. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Operations for Field Elements} The following operations are always available for field elements. Of course the operands must lie in compatible fields, i.e., the fields must be equal, or at least have a common superfield. ' + ' The operator '+' evaluates to the sum of the two field elements and , which must lie in compatible fields. ' - ' The operator '-' evaluates to the difference of the two field elements and , which must lie in compatible fields. ' \*\ ' The operator '\*' evaluates to the product of the two field elements and , which must lie in compatible fields. ' / ' The operator '/' evaluates to the quotient of the two field elements and , which must lie in compatible fields. If the divisor is 0 an error is signalled. ' \^\ ' The operator '\^' evaluates to the -th power of the field element . If is a positive integer then '\^' is '\*\*..\*' ( factors). If is a negative integer '\^' is defined as $1 / {^{-}}$. If 0 is raised to a negative power an error is signalled. Any field element, even 0, raised to the 0-th power yields 1. For the precedence of the operators see "Operations". %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{GaloisGroup} \index{Galois group!of a field} \index{automorphism group!of a field} 'GaloisGroup( )' 'GaloisGroup' returns the Galois group of the field as a group (see "Groups") of field automorphisms (see "Field Homomorphisms"). The Galois group of a field over a subfield '.field' is the group of automorphisms of that leave the subfield '.field' fixed. This group can be interpreted as a permutation group permuting the zeroes of the characteristic polynomial of a primitive element of . The degree of this group is equal to the number of zeroes, i.e., to the dimension of as a vector space over the subfield '.field'. It operates transitively on those zeroes. The normal divisors of the Galois group correspond to the subfields between '' and '.field'. | gap> G := GaloisGroup( GF(4096)/GF(4) );; gap> Size( G ); 6 gap> IsCyclic( G ); true # the Galois group of every finite field is # generated by the Frobenius automorphism gap> H := GaloisGroup( CF(60) );; gap> Size( H ); 16 gap> IsAbelian( H ); true | The default function 'FieldOps.GaloisGroup' just raises an error, since there is no general method to compute the Galois group of a field. This default function is overlaid by more specific functions for special types of domains (see "Field Functions for Finite Fields" and "GaloisGroup for Number Fields"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{MinPol} \index{minimal polynom!of a field element} 'MinPol( )' \\ 'MinPol( , )' In the first form 'MinPol' returns the coefficients of the minimal polynomial of the element in its default field over its prime field (see "DefaultField"). In the second form 'MinPol' returns the coefficients of the minimal polynomial of the element in the field over the subfield '.field'. Let $F/S$ be a field extension and $L$ a minimal normal extension of $S$, containing $F$. The *minimal polynomial* of $z$ in $F$ over $S$ is the squarefree polynomial whose roots are precisely the conjugates of $z$ in $L$ (see "Conjugates"). Because the set of conjugates is fixed under the Galois group of $L$ over $S$ (see "GaloisGroup"), so is the polynomial. Thus all the coefficients of the minimal polynomial lie in $S$. | gap> MinPol( Z(2^6) ); [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ] gap> MinPol( GF(2^12), Z(2^6) ); [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ] gap> MinPol( GF(2^12)/GF(2^2), Z(2^6) ); [ Z(2^2), Z(2)^0, Z(2)^0, Z(2)^0 ] | The default function 'FieldOps.MinPol', which works only for extensions with abelian Galois group, multiplies the linear factors $x - c$ with ranging over the set of conjugates of in (see "Conjugates"). For generic algebraic extensions, it is overlayed by solving a system of linear equations, given by the coefficients of powers of in respect to a given base. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CharPol} \index{characteristic polynom!of a field element} 'CharPol( )' \\ 'CharPol( , )' In the first form 'CharPol' returns the coefficients of the characteristic polynomial of the element in its default field over its prime field (see "DefaultField"). In the second form 'CharPol' returns the coefficients of the characteristic polynomial of the element in the field over the subfield '.field'. The characteristic polynomial is returned as a list of coefficients, the -th entry is the coefficient of $x^{i-1}$. The *characteristic polynomial* of an element $z$ in a field $F$ over a subfield $S$ is the $\frac{[F\:S]}{{\rm deg } \mu}$-th power of $\mu$, where $\mu$ denotes the minimal polynomial of $z$ in $F$ over $S$. It is fixed under the Galois group of the normal closure of $F$. Thus all the coefficients of the characteristic polynomial lie in $S$. The constant term is $(-1)^{F.degree/S.degree}=(-1)^{[F\:S]}$ times the norm of $z$ (see "Norm"), and the coefficient of the second highest degree term is the negative of the trace of $z$ (see "Trace"). The roots (including their multiplicities) in $F$ of the characteristic polynomial of in $F$ are the conjugates (see "Conjugates") of $z$ in $F$. | gap> CharPol( Z(2^6) ); [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ] gap> CharPol( GF(2^12), Z(2^6) ); [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] gap> CharPol( GF(2^12)/GF(2^2), Z(2^6) ); [ Z(2^2)^2, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ] | The default function 'FieldOps.CharPol' multiplies the linear factors $x - c$ with ranging over the conjugates of in (see "Conjugates"). For nonabelian extensions, it is overlayed by a function, which computes the appropriate power of the minimal polynomial. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Norm} \index{norm!of a field element} 'Norm( )' \\ 'Norm( , )' In the first form 'Norm' returns the norm of the field element in its default field over its prime field (see "DefaultField"). In the second form 'Norm' returns the norm of in the field over the subfield '.field'. The *norm* of an element $z$ in a field $F$ over a subfield $S$ is $(-1)^{F.degree/S.degree}=(-1)^{[F\:S]}$ times the constant term of the characteristic polynomial of $z$ (see "CharPol"). Thus the norm lies in $S$. The norm is the product of all conjugates of $z$ in the normal closure of $F$ over $S$ (see "Conjugates"). | gap> Norm( Z(2^6) ); Z(2)^0 gap> Norm( GF(2^12), Z(2^6) ); Z(2)^0 gap> Norm( GF(2^12)/GF(2^2), Z(2^6) ); Z(2^2)^2 | The default function 'FieldOps.Norm' multiplies the conjugates of in (see "Conjugates"). For nonabelian extensions, it is overlayed by a function, which obtains the norm from the characteristic polynomial. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Trace} \index{trace!of a field element} 'Trace( )' \\ 'Trace( , )' In the first form 'Trace' returns the trace of the field element in its default field over its prime field (see "DefaultField"). In the second form 'Trace' returns the trace of the element in the field over the subfield '.field'. The *trace* of an element $z$ in a field $F$ over a subfield $S$ is the negative of the coefficient of the second highest degree term of the characteristic polynomial of $z$ (see "CharPol"). Thus the trace lies in $S$. The trace is the sum over all conjugates of $z$ in the normal closure of $F$ over $S$ (see "Conjugates"). | gap> Trace( Z(2^6) ); 0*Z(2) gap> Trace( GF(2^12), Z(2^6) ); 0*Z(2) gap> Trace( GF(2^12)/GF(2^2), Z(2^6) ); 0*Z(2) | The default function 'FieldOps.Trace' adds the conjugates of in (see "Conjugates"). For nonabelian extensions, this is overlayed by a function, which obtains the trace from the characteristic polynomial. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Conjugates} \index{Galois conjugates!of a field element} \index{conjugates!of a field element, Galois} 'Conjugates( )' \\ 'Conjugates( , )' In the first form 'Conjugates' returns the list of conjugates of the field element in its default field over its prime field (see "DefaultField"). In the second form 'Conjugates' returns the list of conjugates of the field element in the field over the subfield '.field'. In either case the list may contain duplicates if lies in a proper subfield of its default field, respectively of . The *conjugates* of an element $z$ in a field $F$ over a subfield $S$ are the roots in $F$ of the characteristic polynomial of $z$ in $F$ (see "CharPol"). If $F$ is a normal extension of $S$, then the conjugates of $z$ are the images of $z$ under all elements of the Galois group of $F$ over $S$ (see "GaloisGroup"), i.e., under those automorphisms of $F$ that leave $S$ fixed. The number of different conjugates of $z$ is given by the degree of the smallest extension of $S$ in which $z$ lies.\\ For a normal extension $F$, 'Norm' (see "Norm") computes the product, 'Trace' (see "Trace") the sum of all conjugates. 'CharPol' (see "CharPol") computes the polynomial that has precisely the conjugates with their corresponding multiplicities as roots, 'MinPol' (see "MinPol") the squarefree polynomial that has precisely the conjugates as roots. | gap> Conjugates( Z(2^6) ); [ Z(2^6), Z(2^6)^2, Z(2^6)^4, Z(2^6)^8, Z(2^6)^16, Z(2^6)^32 ] gap> Conjugates( GF(2^12), Z(2^6) ); [ Z(2^6), Z(2^6)^2, Z(2^6)^4, Z(2^6)^8, Z(2^6)^16, Z(2^6)^32, Z(2^6), Z(2^6)^2, Z(2^6)^4, Z(2^6)^8, Z(2^6)^16, Z(2^6)^32 ] gap> Conjugates( GF(2^12)/GF(2^2), Z(2^6) ); [ Z(2^6), Z(2^6)^4, Z(2^6)^16, Z(2^6), Z(2^6)^4, Z(2^6)^16 ] | The default function 'FieldOps.Conjugates' applies the automorphisms of the Galois group of (see "GaloisGroup") to and returns the list of images. For nonabelian extensions, this is overlayed by a factorization of the characteristic polynomial. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Field Homomorphisms}% \index{homomorphisms!of fields} Field homomorphisms are an important class of homomorphisms in {\GAP} (see chapter "Homomorphisms"). A *field homomorphism* $\phi$ is a mapping that maps each element of a field $F$, called the source of $\phi$, to an element of another field $G$, called the range of $\phi$, such that for each pair $x,y \in F$ we have $(x+y)^\phi = x^\phi + y^\phi$ and $(xy)^\phi = x^\phi y^\phi$. We also require that $\phi$ maps the one of $F$ to the one of $G$ (that $\phi$ maps the zero of $F$ to the zero of $G$ is implied by the above relations). An Example of a field homomorphism is the Frobinius automorphism of a finite field (see "FrobeniusAutomorphism"). Look under *field homomorphisms* in the index for a list of all available field homomorphisms. Since field homomorphisms are just a special case of homomorphisms, all functions described in chapter "Homomorphisms" are applicable to all field homomorphisms, e.g., the function to test if a homomorphism is a an automorphism (see "IsAutomorphism"). More general, since field homomorphisms are just a special case of mappings all functions described in chapter "Mappings" are also applicable, e.g., the function to compute the image of an element under a homomorphism (see "Image"). The following sections describe the functions that test whether a mapping is a field homomorphism (see "IsFieldHomomorphism"), compute the kernel of a field homomorphism (see "KernelFieldHomomorphism"), and how the general mapping functions are implemented for field homomorphisms. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsFieldHomomorphism}% \index{IsHomomorphism!for fields} 'IsFieldHomomorphism( )' 'IsFieldHomomorphism' returns 'true' if the mapping is a field homomorphism and 'false' otherwise. Signals an error if is a multi valued mapping. A mapping $map$ is a field homomorphism if its source $F$ and range $G$ are both fields and if for each pair of elements $x, y \in F$ we have $(x+y)^{map} = x^{map} + y^{map}$ and $(xy)^{map} = x^{map} y^{map}$. We also require that $1_F^{map} = 1_G$. | gap> f := GF( 16 ); GF(2^4) gap> fun := FrobeniusAutomorphism( f ); FrobeniusAutomorphism( GF(2^4) ) gap> IsFieldHomomorphism( fun ); true | 'IsFieldHomomorphism' first tests if the flag '.isFieldHomomorphism' is bound. If the flag is bound, 'IsFieldHomomorphism' returns its value. Otherwise it calls \\ '.source.operations.IsFieldHomomorphism( )', remembers the returned value in '.isFieldHomomorphism', and returns it. Note that of course all functions that create field homomorphism set the flag '.isFieldHomomorphism' to 'true', so that no function is called for those field homomorphisms. The default function called this way is 'MappingOps.IsFieldHomomorphism'. It computes all the elements of the source of and for each pair of elements $x, y$ tests whether $(x+y)^{map} = x^{map} + y^{map}$ and $(xy)^{map} = x^{map} y^{map}$. Look under *IsHomomorphism* in the index to see for which mappings this function is overlaid. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{KernelFieldHomomorphism}% \index{Kernel!for fields} 'KernelFieldHomomorphism( )' 'KernelFieldHomomorphism' returns the kernel of the field homomorphism . Because the kernel must be a ideal in the source and it can not be the full source (because we require that the one of the source is mapped to the one of the range), it must be the trivial ideal. Therefor the kernel of every field homomorphism is the set containing only the zero of the source. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Mapping Functions for Field Homomorphisms}% \index{IsInjective!for field homomorphisms}% \index{IsSurjective!for field homomorphisms}% \index{equality!for field homomorphisms}% \index{Image!for field homomorphisms}% \index{Images!for field homomorphisms}% \index{PreImage!for field homomorphisms}% \index{PreImages!for field homomorphisms} This section describes how the mapping functions defined in chapter "Mappings" are implemented for field homomorphisms. Those functions not mentioned here are implemented by the default functions described in the respective sections. \vspace{5mm} 'IsInjective( )' Always returns 'true' (see "KernelFieldHomomorphism"). \vspace{5mm} 'IsSurjective( )' The field homomorphism is surjective if the size of the image 'Size( Image( ) )' is equal to the size of the range 'Size( .range )'. \vspace{5mm} ' = ' The two field homomorphism and are are equal if the have the same source and range and if the images of the generators of the source under and are equal. \vspace{5mm} 'Image( )' \\ 'Image( , )' \\ 'Images( , )' The image of a subfield under a field homomorphism is computed by computing the images of a set of generators of the subfield, and the result is the subfield generated by those images. \vspace{5mm} 'PreImage( )' \\ 'PreImage( , )' \\ 'PreImages( , )' The preimages of a subfield under a field homomorphism are computed by computing the preimages of all the generators of the subfield, and the result is the subfield generated by those elements. Look in the index under *IsInjective*, *IsSurjective*, *Image*, *Images*, *PreImage*, *PreImages*, and *equality* to see for which field homomorphisms these functions are overlaid. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Field Records} A field is represented by a record that contains important information about this field. The {\GAP} library predefines some field records, for example 'Rationals' (see "Rationals"). Field constructors construct others, for example 'Field' (see "Field"), and 'GaloisField' (see "GaloisField"). Of course you may also create such a record by hand. All field records contain the components 'isDomain', 'isField', 'char', 'degree', 'generators', 'zero', 'one', 'field', 'base', and 'dimension'. They may also contain the optional components 'isFinite', 'size', 'galoisGroup'. The contents of all components of a field are described below. 'isDomain': \\ is always 'true'. This indicates that is a domain. 'isField': \\ is always 'true'. This indicates that is a field. 'char': \\ is the characteristic of . For finite fields this is always a prime, for infinite fields this is 0. 'degree': \\ is the degree of *as extension of the prime field*, not as extension of the subfield . For finite fields the order of is given by '.char\^ .degree'. 'generators': \\ a list of elements that together generate . That is is the smallest field over the prime field given by '.char' that contains the elements of '.generators'. 'zero': \\ is the additive neutral element of the finite field. 'one': \\ is the multiplicative neutral element of the finite field. 'field': \\ is the subfield over which was constructed. This is either a field record for , or the same value as '.char', denoting the prime field (see "Fields over Subfields"). 'base': \\ is a list of elements of forming a base for as vector space over the subfield . 'dimension': \\ is the dimension of as vector space over the subfield . 'isFinite': \\ if present this is 'true' if the field is finite and 'false' otherwise. 'size': \\ if present this is the size of the field . If is infinite this holds the string \"infinity\". 'galoisGroup': \\ if present this holds the Galois group of (see "GaloisGroup"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E Emacs . . . . . . . . . . . . . . . . . . . . . local Emacs variables %% %% Local Variables: %% mode: outline %% outline-regexp: "\\\\Chapter\\|\\\\Section" %% fill-column: 73 %% eval: (hide-body) %% End: %%