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Math Help - quadratics and polynomials

  1. #1
    Junior Member universalsandbox's Avatar
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    quadratics and polynomials

    What would be all the quadratic polynomials that are irreducible in Z_{3}[x]. Integers mod 3. How would you go about finding them. Similarly, how would you find them for Z_{5}[x].
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    Quote Originally Posted by universalsandbox View Post
    What would be all the quadratic polynomials that are irreducible in Z_{3}[x]. Integers mod 3. How would you go about finding them. Similarly, how would you find them for Z_{5}[x].
    Write out a complete list of polynomials that are quadradic in \mathbb{Z}_3.
    Then see which of them have zeros.
    The ones without zeros are irreducible.
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  3. #3
    Junior Member universalsandbox's Avatar
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    I'm confused on just how to write them out.

    I know that integers mod p, for p prime, gives a field.

    and for F, finite, |F| = p^n.

    In this case, n = quadratic, p=3

    |F| = 3^2 = 9

    and there are p(p+1)/2 = 3(3+1)/2 = 6 reducible ones.

    So there should be 9-6 = 3 irreducible ones. But could someone give me the entire list to get an idea of what a quadratic under Z_{3} is and how to identify them. Thanks.
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    Quote Originally Posted by universalsandbox View Post
    I'm confused on just how to write them out.

    I know that integers mod p, for p prime, gives a field.

    and for F, finite, |F| = p^n.

    In this case, n = quadratic, p=3

    |F| = 3^2 = 9

    and there are p(p+1)/2 = 3(3+1)/2 = 6 reducible ones.

    So there should be 9-6 = 3 irreducible ones. But could someone give me the entire list to get an idea of what a quadratic under Z_{3} is and how to identify them. Thanks.
    The quadradics are:
    Code:
    x^2
    x^2+1
    x^2+2
    x^2+x
    x^2+x+1
    x^2+x+2
    x^2+2x
    x^2+2x+1
    x^2+2x+2
    2x^2
    2x^2+1
    2x^2+2
    2x^2+x
    2x^2+x+1
    2x^2+x+2
    2x^2+2x
    2x^2+2x+1
    2x^2+2x+2
    Now check which ones has zeros in \mathbb{Z}_3.
    The ones without zeros will be irreducible.
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  5. #5
    Junior Member universalsandbox's Avatar
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    If irreducible, then T.
    If NOT irreducible, then F.

    Code:
    F: x^2
    T: x^2+1
    F: x^2+2
    F: x^2+x
    F: x^2+x+1
    T: x^2+x+2
    F: x^2+2x
    F: x^2+2x+1
    T: x^2+2x+2
    F: 2x^2
    F: 2x^2+1
    T: 2x^2+2
    F: 2x^2+x
    T: 2x^2+x+1
    F: 2x^2+x+2
    F: 2x^2+2x
    T: 2x^2+2x+1
    F: 2x^2+2x+2
    Last edited by universalsandbox; December 10th 2008 at 05:51 AM. Reason: revised
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  6. #6
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    You made three mistakes: x^2+x+1,2x^2+1,2x^2+x+2.
    You labeled them as irreducible, while they happen to be reducible.
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  7. #7
    Junior Member universalsandbox's Avatar
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    yep, revised. Thanks.
    Last edited by universalsandbox; December 10th 2008 at 05:51 AM. Reason: revised
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    X^{2}+2 is also reducible.
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