A finite field, as an algebraic set: an ETCS set X whose carrier X::Domain is an axiomatic field with the carrier-author finite-cardinality opt-in. More...

Concept definition

template<typename X, typename Add std::plus<typename X::Domain>, typename Mult std::multiplies<typename X::Domain>>
concept dedekind::algebra::IsGaloisField = 
    IsAlgebraOnSet<X, Add, Mult> &&
    dedekind::category::IsField<typename X::Domain, Add, Mult> &&
    is_galois_field_v<typename X::Domain, Add, Mult>

Detailed Description

A finite field, as an algebraic set: an ETCS set X whose carrier X::Domain is an axiomatic field with the carrier-author finite-cardinality opt-in.

Set-indexed (like the rest of the algebra tower), so it sits at the algebraic-set layer. Uses the axiomatic dedekind::category::IsField on X::Domain so the concept holds on carriers without the division-operator surface (e.g.\ bool under (std::bit_xor, std::bit_and)), which genuinely is a Galois field but does not expose operator/ or .inverse(). Finiteness is the carrier-author opt-in is_galois_field_v; a cardinality-derived pin (via X::cardinality_type) is a follow-up once the cardinality layer is consolidated. Downstream code needing division composes separately with IsDivisionRing.

Template Parameters

X	The set object (`category::IsSet`).
Add	Additive op on `X::Domain` (defaults to `std::plus`).
Mult	Multiplicative op on `X::Domain` (defaults to `std::multiplies`).