Polynomials generating the same $p$-adic fields

Question

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?

Answer 1

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 $\mathbb Q_2$.

Answer 2

Check out Lang's Algebraic Number Theory section II.2. In a nutshell, if two polynomials are $p$-adically close then their roots are close as well, and by Krasner's lemma the fields they generate over $\mathbb Q_p$ will be the same.