Proof that if p is prime, then 3√p is irrational

Proof.Assume not. Assume that given a prime p, 3

√p is rational.

By the Unique Prime Factorization Theorem, we have that we can write

3√p = qa11 ∗ q

a22 ∗ ... ∗ q

ann

where qi is prime ∀i, and ai ∈ Z ∀i

=⇒ ( 3√p)3 = q3a11 ∗ q3a22 ∗ ... ∗ q3ann

=⇒ p1 = q3a11 ∗ q3a22 ∗ ... ∗ q3ann

Since p is a prime, by the Unique Prime Factorization Theorem, we havethat p1 = q3aii for some i. But that means that 1 = 3ai for some i, whereai ∈ Z. So ai =

13 , but ai ∈ Z.

Contradiction. Therefore, the assumption that 3√p is rational is false.

=⇒ that if p is prime, 3√p is irrational.

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