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Math Help - Congruence uncertainty

  1. #1
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    Congruence uncertainty

    Suppose f(x)and g(x)∈Z_7 [x]. Suppose also that f(x)≡g(x)(mod x). What can we say about f(0)and g(0)?

    With this problem i am jus not sure if i am over thinking it because we have done congruences with ideals and in F[x].. would fall under just dealing with F[x]
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  2. #2
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    Quote Originally Posted by empressA88 View Post
    Suppose f(x)and g(x)∈Z_7 [x]. Suppose also that f(x)≡g(x)(mod x). What can we say about f(0)and g(0)?

    With this problem i am jus not sure if i am over thinking it because we have done congruences with ideals and in F[x].. would fall under just dealing with F[x]
    f and g are 7 th degree polynomials in in x. Saying that they are congruent "mod x" f- g is a multiple x which, in turn, means f- g has no constant term.
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    is that f - g, or are you say f and g?
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by HallsofIvy View Post
    f and g are 7 th degree polynomials in in x. Saying that they are congruent "mod x" f- g is a multiple x which, in turn, means f- g has no constant term.
    Doesn't  f,g\in\mathbb{Z}_7[x] mean  f and  g are polynomials with coefficients in  \mathbb{Z}_7 ?

    If so then  f\equiv g\bmod{x} \implies x\mid f-g\implies the constant term of  f-g\equiv0\bmod{7} .

    But this constant term is equal to  f(0)-g(0) . Therefore  f(0)\equiv g(0) \bmod{7} .
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