Zp is an integral domain

Dependencies:

1. Integral Domain
2. Zn is a ring
3. Modular Equivalence
4. Euclid's lemma

If $p$ is prime, $\mathbb{Z}_p$ is an integral domain.

Proof

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

\begin{align} & ab \equiv 0 \pmod{p} \\ &\Rightarrow p \mid ab \\ &\Rightarrow p \mid a \vee p \mid b \tag{by Euclid's lemma} \\ &\Rightarrow a \equiv 0 \pmod{p} \vee b \equiv 0 \pmod{p} \end{align}

Therefore, $a$ and $b$ are not zero-divisors.

Since $\mathbb{Z}_p$ is a commutative ring with no zero-divisors, it is an integral domain.

Dependency for:

1. Gauss' Lemma

Info:

• Depth: 3
• Number of transitive dependencies: 10

Transitive dependencies:

1. Group
2. Ring
3. Integral Domain
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. Euclid's lemma