I'm looking for some criteria for the irreducibility of polynomials with coefficients in $\mathbb{Q}_p((X))$.
In particular, is the polynomial $Y^2+1$ irreducible over $\mathbb{Q}_3((X))$? And how about in $\mathbb{Q}_p((X))$?
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$\begingroup$ So in the particular case, you are really asking when $\mathbb{Q}_p((X))$ has an element that squares to $-1$. $\endgroup$– Tobias KildetoftDec 9, 2013 at 12:06
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$\begingroup$ I know. Maybe I'm being completely obtuse here, but is there a trivial way to know the answer to that? $\endgroup$– user42461Dec 9, 2013 at 12:12
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$\begingroup$ I don't actually know. I just felt that it was worth pointing out that the specific question is probably a lot easier than the general one (since it is indeed a very special case). $\endgroup$– Tobias KildetoftDec 9, 2013 at 12:14
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2 Answers
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The ring $R=\mathbb Q_p[[X]]$ of formal power series is factorial, and its quotient field is your field $S=\mathbb Q_p((X))$ of coefficients. So the Gauss Lemma holds. In particular, if a polynomial $f(Y)\in\mathbb Q_p[Y]$ factorizes over $S$, then it factorizes over $R$, and upon setting $X=0$ it factorizes over $\mathbb Q_p$. So you are asking if $-1$ is a square in $\mathbb Q_p$, and the answer to that is well known.
answered Dec 9, 2013 at 12:46
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Since you ask about $Y^2+1$ over ${\mathbf Q}_p((X))$ and Peter Mueller's answer reduces this to the case of $Y^2+1$ over ${\mathbf Q}_p$, the following broader context might be conceptually useful: if $L/K$ is an extension of fields such that $K$ is algebraically closed in $L$ (that is, any element of $L$ algebraic over $K$ is in $K$), then a polynomial in $K[Y]$ is irreducible in $K[Y]$ if and only if it is irreducible in $L[Y]$. Trivially, irreducibility in $L[Y]$ implies irreducibility in $K[Y]$. The other direction is more interesting and is left as an exercise, along with the task of showing $F$ is algebraically closed in $F((X))$. Thus for any $c \in {\mathbf Q}_p$, irreducibility of $Y^2-c$ over ${\mathbf Q}_p((X))$ is equivalent to irreducibility of $Y^2-c$ over ${\mathbf Q}_p$.
