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Math Help - abstract algebra:help meto find a solution( as soon as possible)

  1. #1
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    abstract algebra:help meto find a solution( as soon as possible)

    let n=p1^r1.p2^r2.p3^r3...pk^rk where p1,p2,..pk are distinct primes and each ri>=0. prove that n is a perfect square if and only if each ri is even.
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  2. #2
    GJA
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    Re: abstract algebra:help meto find a solution( as soon as possible)

    Hi, lenin1234007.

    If each exponent is even then you can write

    n=(p_{1}^{l_{1}}\ldots p_{k}^{l_{k}})^{2}

    Where l_{1}=r_{1}/2, \ldots, l_{k}=r_{k}/2.

    If n is a perfect square, then

    n=t^{2}

    This implies

    p_{1}^{r_{1}}\cdots p_{k}^{r_{k}}=t^{2}

    Now t has its own unique factorization into primes. By the Fundamental Theorem of Arithmetic, the primes and their exponents must be equal on both sides of the equation. Since the exponents of all the primes on the right are even after you square, all the exponents on the left must be even also.

    Does this help? Let me know if something is unclear.

    Good luck!
    Last edited by GJA; September 4th 2012 at 10:51 PM.
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