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Given two polynomials $t(x) = x - r_1$ and $p(x) = x - r_2$ s.t. $r_1 \neq r_2 $ and $r_1, r_2 \in F$ where $F$ is a (finite) field.

Are the polynomials $t(x)$ and $p(x)$ pairwise coprime and why?


Given that $\textbf{GCD(a, b)}$ is defined only if $a,b$ are elements from a Euclidean domain, and since the polynomials above are from a polynomial ring which is in itself a Euclidean domain, what would a "unit" element mean in this context?

Is a polynomial ring with coefficients in a finite field, a finite field?

See: https://proofwiki.org/wiki/Definition:Coprime/Euclidean_Domain

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The polynomial ring $F[x]$ over a (finite) field is indeed a Euclidean domain, and is not a field, as no polynomials of degree $>1$ can have an inverse (because $\deg(fg)=\deg f+\deg g$, so the degree can only increase by multiplying).

This also answers your first questions. The units in $F[x]$ are exactly the nonzero constant polynomials.

Consequently, $x-r_1$ and $x-r_2$ are coprime if $r_1\ne r_2$.

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