Abstract algebra/Fields

Definition: A field is a non empty set $F$ with two binary operations $+$ and $\cdot$ such that (F,$+$,$\cdot$) has commutative unitary ring structure and satisfy the following property:

$\mathrm{\forall }x\in F-\{0\},\mathrm{\exists }y\in F:x\cdot y=1$ (every element in $F$ except for 0 has a multiplicative inverse)

Examples:

1.- $\mathbb{Q},ℝ,ℂ$ (rational, real and complex numbers) with standard $+$ and $\cdot$ operations have field structure. These are fields examples with infinite cardinality.

2.- ${\mathbb{Z}}_p$, the integer set modulo p (p a prime positive integer number) with ${+}_{(modp)}$ and ${\cdot }_{(modp)}$ operations is a family of finite fields.

Exercises

1. Prove that the above examples are fields.