Since I'm new into this type of fields, excuse my lack of knowledge. I'm ambitious and want to learn more about it.
My problem: How can I show, f(x)= $x^2+6$ is irreducible over $\mathbb{Q}_2$ ?
It's easy to show over $\mathbb{Q}$, but how about a rational 2-adic field?
I think first I have to look at it in 2-adic integer Ring $\mathbb{Z}_2$?