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Math Help - Proving prime is irrational

  1. #1
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    Proving prime is irrational

    Heres a proof I am getting stuck with

    Prove sqrt(p) is irrational, p is set of all primes

    so heres what Ihave

    prove sqrt(p) is rational
    sqrt(p) = a/b a,b set of integers, b not equal to 0, and a/b is in lowest form

    p = a^2/b^2
    p*b^2 = a^2.

    a^2/p = b^2.

    a/sqrt(P) = b

    Kind of at a standstill here, since b cannot be rational when dividing by the sqrt of a prime. how can I say this?
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    If gcd(a,b)=1 then also gcd(a^2,b^2)=1.

    You got: a^2|pb^2 ==> a^2|p ==>a=1 ==>1=pb^2(contradiction!)


    read here to understand the first move: Euclid's lemma - Wikipedia, the free encyclopedia
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