Field

Symbol-free definition
A field is a commutative unital ring with the additional property that its multiplicative group comprises all the nonzero elements, that is, with the property that all nonzero elements are invertible.

Alternatively, a field is a commutative unital ring with no proper nontrivial ideal.

Definition with symbols
A field is a set $$F$$ endowed with constants $$0$$ and $$1$$ (not equal), a unary operation $$-$$ and binary operations $$+$$ and $$*$$ such that:


 * $$a + (b + c) = (a + b) + c$$ for all $$a, b, c$$ in $$F$$
 * $$a + 0 = a$$ for all $$a$$ in $$F$$
 * $$a + b = b + a$$ for all $$a, b$$ in $$F$$
 * $$a + (-a) = 0$$ for all $$a$$ in $$F$$
 * $$a * (b * c) = (a * b) * c$$ for all $$a, b, c$$ in $$F$$
 * $$a * 1 = a$$ for all $$a$$ in $$F$$
 * $$a * b = b * a$$ for all $$a, b$$ in $$F$$
 * $$a * (b + c) = (a * b) + (a * c)$$ for all $$a, b, c$$ in $$F$$
 * For all nonzero $$a$$ in $$F$$, there exists a $$b$$ in $$F$$ such that $$a * b = 1$$

Metaproperties
A subring of a field need not be a field. It could be any integral domain. For instance $$\mathbb{Z}$$ is a subring of $$\R$$.

Module theory
Modules over fields are precisely the same as vector spaces. In particular any finitely generated module over a field is a free module and the number of generators is independent of the choice of generating set.