6.17 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 F are described below.

`isDomain`:

is always `true`. This indicates that F is a domain.

`isField`:

is always `true`. This indicates that F is a field.

`char`:

is the characteristic of F. For finite fields this is always a prime, for infinite fields this is 0.

`degree`:

is the degree of F as extension of the prime field, not as extension of the subfield S. For finite fields the order of F is given by `F.char^ F.degree`.

`generators`:

a list of elements that together generate F. That is F is the smallest field over the prime field given by `F.char` that contains the elements of `F.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 S over which F was constructed. This is either a field record for S, or the same value as `F.char`, denoting the prime field (see Fields over Subfields).

`base`:

is a list of elements of F forming a base for F as vector space over the subfield S.

`dimension`:

is the dimension of F as vector space over the subfield S.

`isFinite`:

if present this is `true` if the field F is finite and `false` otherwise.

`size`:

if present this is the size of the field F. If F is infinite this holds the string "infinity".

`galoisGroup`:

if present this holds the Galois group of F (see GaloisGroup).

GAP 3.4.4
April 1997