"The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits." G.K. Chesterton

Friday, June 18, 2010

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.

Euclid's first theorem states that for a, bZ and any prime p:
p|ab  →   p|a  or  p|b 
An immediate corollary of it is, for a, 1 ≤ nZ and any prime p:
p|an   →   p|a
p|an = aan-1    →  p|a  or  p|an-1
p|an-1 = aan-2   →   p|a  or  p|an-2
p|an-(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 (p1/n = a / b), where a nad b have no common factor.
p1/n = a / b   →   p =  an / bn 
→   pbn =  an  →   p|an   by def. of divides, since bnZ
→  p|a by Corollary to Euclid's first theorem.
Hence a = pk ,  k∈ Z by def. of p|a.        (1)
But  p1/n = a / b   →   a = p1/nb = pk   →   pbn = pnkn 
→ bn = p( pn-2kn )  →   p|bn   by def. of divides, since pn-2knZ
→ 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 (p1/n ≠ a / b)  q.e.d.

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