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Math Help - abstract algebra #3

  1. #1
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    abstract algebra #3

    PLease help me with this:

    If p is prime and p|a^n, is it true that p^n|a^n? Justify your answer.

    Thank you.
    Last edited by mr fantastic; April 15th 2010 at 04:37 PM. Reason: Edited post title
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  2. #2
    Senior Member Dinkydoe's Avatar
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    Yes indeed. If p|a^n \Rightarrow p|a. You can show this inductively:

    It follows from the prime-property: p|xy \Rightarrow p|x or p|y. Take x= a, y= a^{n-1}. If p|a you're done if p|a^{n-1} you go on with x= a, y= a^{n-2}. Eventually you'll obtain p|a

    Conclusion: p^n|a^n
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  3. #3
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    Quote Originally Posted by Dinkydoe View Post
    Yes indeed. If p|a^n \Rightarrow p|a. You can show this inductively:

    It follows from the prime-property: p|xy \Rightarrow p|x or p|y. Take x= a, y= a^{n-1}. If p|a you're done if p|a^{n-1} you go on with x= a, y= a^{n-2}. Eventually you'll obtain p|a

    Conclusion: p^n|a^n
    thank you. you showed p|a. but what about p^n.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Deepu View Post
    thank you. you showed p|a. but what about p^n.
    p\mid a\implies a=kp so a^n=k^np^n...so
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