# 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?

## 2 Answers

The ideas about "Newton polygons" subsume Eisenstein's criterion, and are set up to be applied in a straightforward way to polynomials over Dedekind rings, e.g., number fields. For given ideal in the base ring, graph the convex-upward hull of the ords of the coefficients at that ideal. The (negative) slopes of the resulting line segments are the ords of the roots, with respect to the extension(s) of the ord on the groundfield, with multiplicities (=the length of the segment with given slope). Thus, if there is a single line segment with slope $m/n$ (in lowest terms) and $n$ is the degree, the polynomial is irreducible. A straightforward argument (modulo basic algebraic number theory) is visible on-line at http://www.math.umn.edu/~garrett/m/number_theory/newton_polygon.pdf

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).