# 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:

• $k$ forms an 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:

$a * (b + c) = (a * b) + (a * c)$

• $k^* = k \setminus 0$ forms an abelian group with binary operation $*$, inverse map ${}^{-1}$ and identity element $1$. This is called the multiplicative group of $k$.