$f(T)=T^2 + 1$ irreducibility over $p$-adic number polynomials, $\mathbb Q_p[T]$.

Question

In relation to this previous post, Why is $x^2+1$ irreducible in $\mathbb{Q}_2[x]$., is there something we can say about $f(T) = T^2+1$ in generality over different choices of $p$ of $\mathbb Q_p[T]$? Can someone point me toward the general statement of what is said in https://math.stackexchange.com/a/2260981/1030758.

Answer 1

Since $T^2+1$ has distinct roots in an algebraic closed field of characteristic $\neq 2$ it will have a root in $\mathbb{Q}_p$ if and only if it has a root modulo $p$, e.g. by Hensel's Lemma, when $p\neq2$.

Now $T^2+1$ has a root modulo $p$ if and only if $-1$ is a square modulo $p$ and the latter is true if and only if $p\equiv1\bmod 4$ (of which there are infinitely many by Dirichlet's Theorem).

For instance, $T^2+1$ has two roots $z_1$, $z_2$ in $\mathbb{Q}_5$ and we know that $$ |z_1-2|_5<1,\qquad |z_2-3|_5<1 $$ since $2^2\equiv3^2\equiv-1\bmod5$. A better approximation will result from computing the square roots of $-1$ modulo $25$, modulo $125$, modulo $625$ and so forth.