Field

Definition
A field is a set (that we'll call $$k$$) equipped with the following:


 * An (infix) binary operation $$+$$, called the addition
 * An (infix) binary operation $$*$$, called the multiplication
 * Two distinct constants $$0, 1 \in k$$, called zero and one
 * A unary operation denoted by the prefix symbol $$-$$ (called the negative, or additive inverse)
 * A map $${}^{-1}: k \setminus \{ 0 \} \to k \setminus \{ 0 \}$$

such that the following compatibility conditions hold:

$$a * (b + c) = (a * b) + (a * c)$$
 * $$k$$ forms an defining ingredient::abelian group with binary operation $$+$$, inverse map $$-$$, and identity element $$0$$. This is called the additive group of $$k$$.
 * $$*$$ gives an associative binary operation on $$k$$
 * The following distributivity law holds:
 * $$k^* = k \setminus 0$$ forms an abelian group with binary operation $$*$$, inverse map $${}^{-1}$$ and identity element $$1$$. This is called the multiplicative group of $$k$$.