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I was doing this question as a homework from a lecture. I have proven that for all prime numbers, the square root is irrational using Euler’s lemma. However, I’m stuck on how to prove it for all non perfect squares. How should I approach this problem? Any help or solutions would be appreciated! Thank you so much!

Bill Dubuque
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1937874
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  • Is $1/4$ considered a "non perfect square"? – peterwhy Mar 11 '24 at 04:12
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    There are lots of other answers to the same question on this site, besides the one I linked. Please use the search function on this site, or better yet, use google with “site:math.stackexchange.com” added to your query. – Aig Mar 11 '24 at 04:20

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Let $n$ be a non-perfect square. Then by definition if $n = p_1^{\alpha_1}\cdots p_k^{\alpha_k}$ is the prime factorization of $n$, then at least one of the $\alpha$'s is odd. Let's say odd alphas are $\alpha_{k_1},\cdots,\alpha_{k_m}$. Then $n$ can be written as $n = A^2 \cdot \prod_{j = 1}^m p_{k_j}$ for some $A\in\mathbb{Z}$. So $\sqrt{n}$ is irrational $\iff \sqrt{\prod_{j = 1}^m p_{k_j}}$ is irrational $\iff \prod_{j = 1}^m p_{k_j} = \frac{a^2}{b^2}$ for $a,b \in\mathbb{Z}$ that are relatively prime, which is a contradiction.

mathemagician
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  • Please strive not to post more (dupe) answers to dupes of FAQs. This is enforced site policy, see here. It's best for site health to delete this answer (which also minimizes community time wasted on dupe processing.) – Bill Dubuque Mar 11 '24 at 04:40
  • Can you explain how does n= A^2 . pi(pkj)? – 1937874 Mar 11 '24 at 08:34
  • @1937874: When you do the prime factorization of a number, some of the alpha exponents will be odd, some will be even (e.g., 72 = 2^3 * 3^2, the exponent of 2 is odd, the exponent of 3 is even). Odd exponents (i.e., 2k+1) can be written as the product of the prime and a square. (e.g.. 72 = 2^3 * 3^2 = 22^2 3^2 = (23)^2 2 = 6^2 * 2) – mathemagician Mar 13 '24 at 02:25
  • Sorry to ask again, but why is pi(pkj)=a^2/b^2? – 1937874 Mar 14 '24 at 03:26
  • @1937874 - If the square root is rational (assume for the sake of contradiction), then the squared value of the square root can be written as a^2/b^2. – mathemagician Mar 14 '24 at 03:30
  • But is it rational? – 1937874 Mar 14 '24 at 03:44
  • How can we prove that to be rational? – 1937874 Mar 14 '24 at 03:44
  • Kind of understand. – 1937874 Mar 14 '24 at 04:15