Checking irreducibility of polynomials over number fields Are there general methods for checking irreducibility of polynomials over number fields? For instance, letting $F = \mathbb{Q}(\sqrt{3})$, I want to know whether $x^3 - 10 + 6\sqrt{3}$ is irreducible over $F$ (I know that it is but it's not trivial). There is the generalized Eisenstein criterion but it's unwieldy when you're not working over the rationals. Are there any methods as useful, or almost as useful, as Eisenstein's criterion, reducing modulo a prime, etc?

The "usual" algorithm for factoring polynomials over number fields go back to Kronecker, and uses the idea that it is essentially sufficient to factor the norm. for a bit of a discussion, see my recent preprint (which is really about something else, but gives a description when talking about an algorithm to compute the Galois group), and references therein (e.g. to Susan Landau's original paper).
