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Let $K[t]$ be the ring of polynomials over a field $K$. Let $K(t)$ be its fraction field. Let $f$ be an irreducible polynomial in $K(t)[x]$. There exists an element $a\in K[t] $ such that $af$ is in $K[t,x]$.

Does there exist $a \in K[t]$ such that $a f$ is irreducible in $K[t,x]$?

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    You are almost right : to get$a$primitive polynomial, you have to multiply by the l.c.m. of the denominators of the coefficients AND to divide by the g.c.d. of their numerators.2012-11-05

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