N>1 roots of primes are irrational

The question came up in a conversation recently, and although it seemed intuitively true, I couldn't find a proof online. Many of you reading this are probably familar with the proof for the irrationality of √2. Well, this is a more general version for all n integer roots of any prime.

Theorem

Proof

Euclid's first theorem states that for a, b ∈ Z and any prime p:

p|ab  →   p|a  or  p|b 

An immediate corollary of it is, for a, 1 ≤ n ∈ Z and any prime p:

p|aⁿ   →   p|a

Since

p|aⁿ = aa^n-1    →  p|a  or  p|a^n-1

but,

p|a^n-1 = aa^n-2   →   p|a  or  p|a^n-2

:

p|a^n-(n-2) = a² = aa   →   p|a  or  p|a

→  p|a  or  p|a  or  p|a  or... p|a  ( n times )   →   p|a

* * *

Now suppose for contradiction that

∃a,b,2≤n∈ Z, p∈P (p^1/n = a / b), where a nad b have no common factor.

p^1/n = a / b   →   p =  aⁿ / bⁿ 

→   pbⁿ =  aⁿ  →   p|aⁿ   by def. of divides, since bⁿ∈ Z

→  p|a by Corollary to Euclid's first theorem.

Hence a = pk ,  k∈ Z by def. of p|a.        (1)

But  p^1/n = a / b   →   a = p^1/nb = pk   →   pbⁿ = pⁿkⁿ 

→ bⁿ = p( p^n-2kⁿ )  →   p|bⁿ   by def. of divides, since p^n-2kⁿ∈ Z

→ p|b by Corollary to Euclid's first theorem.

Hence b = pm ,  m∈ Z by def. of p|b.     (2)

Statements (1) and (2) jointly contradict the statement that "a and b have no common factor".

Hence ∀a,b,2≤n∈ Z, p∈P (p^1/n ≠ a / b)  q.e.d.