Group gradings on M3(k)

We describe and classify all group gradings on the matrix algebra M-3(k), where k is an arbitrary field. We show that any such grading is either isomorphic to a good grading, for which all the matrix units are homogeneous elements, or reduces to a C-3-grading or to a C-3 x C-3-grading. We show that a grading which is not isomorphic to a good grading is a graded division ring. The isomorphism types of non-good C-3-gradings are in a bijective correspondence to cubic Galois extensions of k. The non-good C-3 x C-3-gradings which do not reduce to C-3-gradings are fine gradings, and their description also depends on cubic Galois extensions of k.