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This question originates from Pinter's A Book of Abstract Algebra, Chapter 27, Exercise H1.

Prove if $a(x)$ and $b(x)$ have a common root $c$ in some extension of $F$, they have a common factor of positive degree in $F[x]$. [Use the fact that $a(x), b(x) \in \operatorname{ker} \sigma_c$.]

[Edited]

Let $\sigma_c$ be the substitution function such that $\sigma_c(a(x))=a(c)$. Note the range of $\sigma_c$ is the minimal extension $F(c)$, and $\sigma_c$ is a homomorphism.

Let $E$ be some extension of $F$, so $F(c) \subseteq E$. The kernel of $\sigma_c$ consists of all the polynomials in $F[x]$ such that $c$ is a root. Thus $a(x), b(x)\in \operatorname{ker}\sigma_c$. Since the kernel of any homomorphism is an ideal and every ideal of $F[x]$ is principal, $a(x)$ and $b(x)$ must have a common factor of positive degree in $F[x]$.

Correct?

user26857
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hchar
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1 Answers1

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Here is an alternative solution:

Let $E$ be an extension of $F$ where $a(x)$ and $b(x)$ have a common root $c$. Since $I=\langle x-c \rangle$ is a prime ideal of $E[x]$, we have $J=I\cap F[x]$ is a prime ideal of $F[x]$. Since $F[x]$ is a PID, we have $J=\langle p(x) \rangle$, for some $p$ irreducible. Since $a(x)$ and $b(x)$ are in $I$, they are in $J$, and so they are both multiples of $p(x)$.

lhf
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