Algebraic closure of finite prime fields

It is sometimes useful to consider various finite fields in a fixed characteristic at the same time, together with natural embeddings between these fields. The fields returned by abelian_closure are intended for that purpose.

algebraic_closure — Method

algebraic_closure(F::FinField)

Let F be a prime field of order p. Return a field K that is the union of finite fields of order p^d, for all positive integers d. The degree d extension of F can be obtained as ext_of_degree(K, d).

K is cached in F, and the fields returned by ext_of_degree are cached in K.

Examples

julia> K = algebraic_closure(GF(3, 1));

julia> F2 = ext_of_degree(K, 2);

julia> F3 = ext_of_degree(K, 3);

julia> x = K(gen(F2)) + K(gen(F3));

julia> degree(x)
6

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ext_of_degree — Function

ext_of_degree(A::AlgClosure, d::Int)

Return a finite field F of order p^d where p is the characteristic of K. This field is compatible with A in the sense that A(x) returns the element of A that corresponds to the element x of F.

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