field

Field

Set

context	$X$
definiendum	$\langle X,+,* \rangle \in \mathrm{field}(X)$
postulate	$\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$
postulate	$\langle X,* \rangle \in \mathrm{abelianGroup}(X)$

Discussion

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.

Theorems

Finite fields are completely determined by their cardinality.

There is a field of cardinality for each $p^n$ , $p$ prime.

Reference

Wikipedia: Field

Parents

Subset of

Division ring