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Math Help - Finite Fields

  1. #1
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    Finite Fields

    I need a little help im close but stuck.

    1. Is there a field with 25 elements?

    Well I know that |V_F|=p^n, where p is a prime. Then |V_F|=p^n=25=5^2, p=5, n=2. So now I have to construct the monic irreducible polynomial I know that it is degree 2 but how do you get the correct coefficients? I think the field is related to Z_5/p(x). but im stuck.

    2. If W_F has 49 element, the what are the possibilities for F. Describe the collection of subspaces of W_F for each case.

    thanks for the help.
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  2. #2
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    Quote Originally Posted by GreenandGold View Post
    1. Is there a field with 25 elements?

    Well I know that |V_F|=p^n, where p is a prime. Then |V_F|=p^n=25=5^2, p=5, n=2. So now I have to construct the monic irreducible polynomial I know that it is degree 2 but how do you get the correct coefficients? I think the field is related to Z_5/p(x). but im stuck.
    Remember what was said in your last post about this. Let p(x) = x^2 - 2. Check that this polynomial is irreducible and then construct \mathbb{Z}_5[x]/(x^2 - 2). This would be a field with 25 elements.

    2. If W_F has 49 element, the what are the possibilities for F. Describe the collection of subspaces of W_F for each case.
    What do you mean by this problem?
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  3. #3
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    How would I construct a polynomial in general for say degree n? degree 10 or something....

    2. If V_F has 49 elements, the what are the possibilities for F. Describe the collection of subspaces of V_F for each case.
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  4. #4
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    Quote Originally Posted by GreenandGold View Post
    How would I construct a polynomial in general for say degree n? degree 10 or something....
    There always exists an irreducible polynomial in general for a given degree over a finite field.
    The problem is I do not know of any "nice" ways of finding them. What you are asking is basically, "is there an easy way to know if a number is prime or not?". There is no easy way to know if a number is prime and if you are made to search for primes you just use trail and error. The same with polynomial. You start listing polynomials of degree n and then hoping that you found an irreducible one. But the real problem is determining if a polynomial is irreducible or not. There is no efficient way to do this. So doing this problem by hand is a horrible. There are computer programs that can do it fast. You should find yourself a computer program if you want to actually start finding irreducible polynomials.
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  5. #5
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    Are there any free programs? I dont want to invest in one. Should I just try this and see if it is reduce able over Z_p?



    What about the other question? Are the only possiblities for F just 7 since p^n=7^2, 7 a prime.
    Last edited by GreenandGold; February 10th 2009 at 05:34 PM.
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  6. #6
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    Quote Originally Posted by GreenandGold View Post
    Are there any free programs? I dont want to invest in one. Should I just try this and see if it is reduce able over Z_p?

    What about the other question?
    There are some free program out there.
    Try PARI
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