Let $k$ be a field , $f\in k[x,y]$ such that $f$ is not divisible by any non-constant polynomial of $k[x]$ nor any non-constant polynomial of $k[y]$ . Then is it true that $f$ viewed in $k(x)[y]$ is irreducible iff $f$ viewed in $k(y)[x]$ is irreducible ?
Concerning a polynomial $f\in k[x,y]$ is not divisible by any non-constant polynomial of $k[x]$ nor of $k[y]$
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1 Answers
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Yes. If $f$ is not divisible by any non-constant polynomial of $k[x]$ nor of $k[y]$, then it is "primitive" viewed in $k[x][y]$ and $k[y][x]$ respectively. Since $k[t]$ is a UFD, you can apply Gauss' Lemma two times and get
$f$ is irreducible in $k(y)[x] \Leftrightarrow$ $f$ is irreducible in $k[y][x]=k[x][y]\Leftrightarrow$ $f$ is irreducible in $k(x)[y]$.