TITLE: Polynomials generating the same $p$-adic fields
QUESTION [6 upvotes]: I wonder if the following fact is true:
Pick $l\in \mathbb N$ a number and let $f,g\in \mathbb Z_p[x]$ be monic polynomials with coefficients in the ring of $p$-adic integers such that $f\equiv g \pmod{p^l}$ and they are irreducible mod $p^l$.
Then the roots of $f$ generate the same field as the roots of $g$.
Can someone help me proving this or finding a counterexample?

REPLY [4 votes]: This is false. For example, $X^2+2$ and $X^2+6$ are both equal and irreducible mod $4$. 
However, since $\mathbb Q_2$ does not contain a square root of $3$, their roots give different extensions of $\mathbb Q_2$.