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 $\mathbb{Q}\_p((X))$?



Irreducible polynomials in $\mathbb{Q}_p((X))[Y]$
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.


