How to prove: if a,b∈Na,b \in \mathbb N, then a1/ba^{1/b} is an integer or an irrational number?

It is well known that 2 is irrational, and by modifying the proof (replacing ‘even’ with ‘divisible by 3‘), one can prove that 3 is irrational, as well. On the other hand, clearly n2=n for any positive integer n. It seems that any positive integer has a square root that is either an integer or irrational number.

  1. How do we prove that if aN, then a is an integer or an irrational number?

I also notice that I can modify the proof that 2 is irrational to prove that 32,42, are all irrational. This suggests we can extend the previous result to other radicals.

  1. Can we extend 1? That is, can we show that for any a,bN, a1/b is either an integer or irrational?

Answer

Theorem: If a and b are positive integers, then a1/b is either irrational or an integer.

If a1/b=x/y where y does not divide x, then a=(a1/b)b=xb/yb is not an integer (since yb does not divide xb), giving a contradiction.

I subsequently found a variant of this proof on Wikipedia, under Proof by unique factorization.

The bracketed claim is proved below.

Lemma: If y does not divide x, then yb does not divide xb.

Unique prime factorisation implies that there exists a prime p and positive integer t such that pt divides y while pt does not divide x. Therefore pbt divides yb while pbt does not divide xb (since otherwise pt would divide x). Hence yb does not divide xb.

[OOC: This answer has been through several revisions (some of the comments below might not relate to this version)]

Attribution
Source : Link , Question Author : Community , Answer Author : Cloudscape

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