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Math Help - Abstract Algebra

  1. #1
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    Abstract Algebra

    How can I show that 1 is not a linear combination of polynomials 2, and x in Z[x].... It is a bit easier to think of it as showing that you can't find polynomials f, g in Z[x] where 2f(x) + xg(x) = 1..Thanks.. I don't think the division algorithm is helpful here.
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  2. #2
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    Quote Originally Posted by JohnStaphin View Post
    How can I show that 1 is not a linear combination of polynomials 2, and x in Z[x].... .
    What is the constant term of the polynomial 2f(x) + xg(x)?
    Whatever it is, it must be an even number. But 1 is odd.
    Which is a contradiction.
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  3. #3
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    That might be a good point, but who says there is a constant term...and even if there was...that can't be enough to show that the other terms don't add up to a negative odd number..which could cancel down to 1..
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  4. #4
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    Quote Originally Posted by JohnStaphin View Post
    That might be a good point, but who says there is a constant term...and even if there was...that can't be enough to show that the other terms don't add up to a negative odd number..which could cancel down to 1..
    f(x) = a_0 + a_1 x + ... + a_n x^n
    g(x) = b_0 + b_1 x + ... + b_m x^m

    Then 2f(x) + xg(x) = 2a_0 + (2a_1 + b_0)x + (2a_2 + b_1)x^2 + ...

    But it is impossible for 2a_0 = 1.
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  5. #5
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    oh... >< You're right. Thanks
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