| context | $X$ |
| definiendum | $\langle X,+,* \rangle \in \mathrm{field}(X)$ |
| postulate | $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ |
| postulate | $\langle X,* \rangle \in \mathrm{abelianGroup}(X)$ |
A field is essentially two compatible abelian groups over a set $X$, one of which is necessarily commutative. Compatible in the sense of the distributive laws of a ring, which is asymmetrical with respect to “$+$” and “$*$”.
The second requirement destinguishes the division ring from a division ring by requiring commutivity and of the multiplication $*$.
One generally (also) calls $F$ the field.
Finite fields are completely determined by their cardinality.
There is a field of cardinality for each $p^n$, $p$ prime.
Wikipedia: Field