# Congruence uncertainty

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• Apr 24th 2010, 09:53 AM
empressA88
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]
• Apr 24th 2010, 12:50 PM
HallsofIvy
Quote:

Originally Posted by empressA88
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.
• Apr 24th 2010, 02:15 PM
empressA88
is that f - g, or are you say f and g?
• Apr 24th 2010, 05:52 PM
chiph588@
Quote:

Originally Posted by HallsofIvy
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 $f(0)\equiv g(0) \bmod{7}$.