# Zp is a field

## Dependencies:

Let $p$ be a prime number. Then $\mathbb{Z}_p$ is a field.

## Proof

$\mathbb{Z}_p$ is a commutative ring with unity.

\[ a \in \mathbb{Z}_p - \{0\} \Rightarrow \gcd(a, p) = 1 \Rightarrow \exists x, ax \equiv 1 \pmod{p} \]

Here $x$ is a multiplicative inverse of $a$. Therefore, a multiplicative inverse exists for every element in $\mathbb{Z}_p - \{0\}$. Therefore, $\mathbb{Z}_p$ is a field.

## Dependency for: None

## Info:

- Depth: 3
- Number of transitive dependencies: 10