# 18.10 GaloisField

`GaloisField( p^d )`
`GF( p^d )`
`GaloisField( p|S, d|pol|bas )`
`GF( p|S, d|pol|bas )`

`GaloisField` returns a field record (see Field Records) for a finite field. It takes two arguments. The form `GaloisField(p,d)`, where p,d are integers, can also be given as `GaloisField(p^d)`. `GF` is an abbreviation for `GaloisField`.

The first argument specifies the subfield S over which the new field F is to be taken. It can be a prime or a finite field record. If it is a prime p, the subfield is the prime field of this characteristic. If it is a field record S, the subfield is the field described by this record.

The second argument specifies the extension. It can be an integer, an irreducible polynomial, or a base. If it is an integer d, the new field is constructed as the polynomial extension with the Conway polynomial of degree d over the subfield S. If it is an irreducible polynomial pol, in which case the elements of the list pol must all lie in the subfield S, the new field is constructed as polynomial extension of the subfield S with this polynomial. If it is a base bas, in which case the elements of the list bas must be linear independently over the subfield S, the new field is constructed as a linear vector space over the subfield S.

Note that the subfield over which a field was constructed determines over which field the Galois group, conjugates, norm, trace, minimal polynom, and characteristic polynom are computed (see GaloisGroup, Conjugates, Field Functions for Finite Fields).

```    gap> GF( 2^4 );
GF(2^4)
gap> GF( GF(2^4), 2 );
GF(2^8)/GF(2^4) ```

GAP 3.4.4
April 1997