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:

Transitive dependencies:

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