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Math Help - unit

  1. #1
    Senior Member Sampras's Avatar
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    unit

    Suppose  f(x) is relatively prime to  0_F . Then  \gcd(f(x), 0_F) = 1_F . So  1_F = f(x)u(x)+0_{F}v(x) for some polynomials  u(x) and  v(x) . Thus  1_F = f(x)u(x) .

    So this means that  f(x) is a unit?
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  2. #2
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    Quote Originally Posted by Sampras View Post
    Suppose  f(x) is relatively prime to  0_F .
    this never happens! every polynomial obviously divides the zero polynomial. \

    EDIT: see my next post!
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  3. #3
    Senior Member Sampras's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    this never happens! every polynomial obviously divides the zero polynomial.
    So since the hypothesis is false, what can be said about  f(x) ? It is not a polynomial?
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  4. #4
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    Quote Originally Posted by Sampras View Post
    So since the hypothesis is false, what can be said about  f(x) ? It is not a polynomial?
    ok, i was too quick in responding i guess ... sorry about that! as i said, every polynomial divides the zero polynomial and thus \gcd(0,f(x))=f(x), for any polynomial. so \gcd(0,f(x))=1 if and

    only if f(x) and g(x)=1 are associates, i.e. if and only if f(x) is a unit. so your conclusion is right.
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