Definition 1.1.2: A Field

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 following axioms:

1. Both + and * are associative, i.e. a+(b+c)=(a+b)+c and a*(b*c)=(a*b)*c
2. Both + and * are commutative, i.e. a+b=b+a and a*b=b*a
3. The distributive law holds, i.e. a*(b+c)=(a*b)+(a*c)
4. 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
5. 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

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