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Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$

Suppose $f(x)$ and $g(x)$ are relatively prime in $F[x].$ Then there exist $\mu(x), \lambda(x) \in F[x]$ such that $\mu(x) f(x) + \lambda(x) g(x) = 1.$ Since $F[x] \subset K[x]$ it follows that $\mu(x), \lambda(x) \in K[x]$ also. Thus $\mu(x) f(x) + \lambda(x) g(x) = 1$ in $K[x] \Longrightarrow f(x)$ and $g(x)$ are relatively prime in $K[x]. \Box$

Is this the correct reasoning? Am I missing any details?

user26857
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St Vincent
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1 Answers1

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Suppose $f(x)$ is relatively prime to $g(x)$ in $F[x]$. If $f(x), g(x)$ are not relatively prime in $K[x]$, they must have a common root in some extension $E$ of $K$, so $F\subseteq K\subseteq E$. Therefore, by this proof, $f(x)$ and $g(x)$ have a common factor of positive degree in $F[x]$; a contradiction.

Conversely, if $f(x)$ and $g(x)$ have a common factor in $F[x]$, clearly they have the same common factor in $K[x]$. This proves the contrapositive.

hchar
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