A field is a set F together with two operations commonly denoted as
+ and *, as well as two different special elements
commonly denoted as 0 and 1, that satisfies the
- Both + and * are associative, i.e.
a+(b+c)=(a+b)+c and a*(b*c)=(a*b)*c
- Both + and * are commutative, i.e.
a+b=b+a and a*b=b*a
- The distributive law holds, i.e.
- 0 is the additive identity, and 1 is the
multiplicative identity, i.e. for all x we have
x+0=x and x*1=x
- There are additive and multiplicative inverses, i.e.
for all x exists y such that x+y=0 and
for all non-zero a exists b such that a*b=1