# Algebraic field

A field is a commutative ring \((R, +, \times)\) (henceforth abbreviated simply as \(R\), with multiplicative identity \(1\) and additive identity \(0\)) which additionally has the property that every nonzero element has a multiplicative inverse: for every \(r \in R\) there is \(x \in R\) such that \(xr = rx = 1\). Conventionally we insist that a field must have more than one element: equivalently, \(0 \not = 1\).

# Examples

Children:

- Field homomorphism is trivial or injective
Field homomorphisms preserve a

*lot*of structure; they preserve so much structure that they are always either injective or totally boring.