Zp is a field

Dependencies:

1. Field
2. Zn is a ring
3. Existence of Modular Inverse

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

Transitive dependencies:

1. Group
2. Ring
3. Field
4. Modular Equivalence
5. Modular multiplication
6. Modular addition
7. Integer Division Theorem
8. Zn is a ring
9. GCD is the smallest Linear Combination
10. Existence of Modular Inverse