Zp is a field


  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.


$\mathbb{Z}_p$ 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


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