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I'm studying abstract algebra and ran into the problem of solving equations where solutions are polynomials.

The problem is as follows: Given B a member of a polynomial field $F[x]$, having coefficients in field $F$. Find all the possible polynomials X in $F[x]$ where $X = B$. That is finding the tuples of coefficients that satisfy $X = B$.

What is not clear to me is how one defines the equation $X = B$: $X = B$ for all $x$ in $F$ or for at least one $x$?

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    It's rather bad form to refer to $F[x]$ as a "polynomial field", given that it is *not* a field: it's a ring. Your question also does not make too much sense to me: a "member of $F[x]$" would mean, to me, a polynomial. The *only* polynomial in $F[x]$ that is equal to $B$ would be $B$ itself.2011-11-26
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    Do you mean: given a polynomial $B\in F[x]$, which other polynomials $X\in F[x]$ satisfy $B(a)=X(a)$ for all $a\in F$? Or, in other words, which polynomials define the same function from $F$ to $F$?2011-11-26

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