The binary tetrahedral group as a p-adic Galois group.

Let be the binary tetrahedral group. This group appears as the double cover of the group of rotations of the tetrahedron (under ), as a group of units in an appropriate -form of the quaternion group, or as .

Consider a local field of residue characteristic . Now consider the Galois group of a finite Galois extension. It has a large pro- part, together with two cyclic parts corresponding to the tamely ramified part and the unramified part. This structure alone shows that cannot be such a group unless .

Alternatively, every 2-dimensional irreducible representation of the Galois group of a local field of odd residue characteristic is induced, and has no index two subgroups, so again cannot occur as a Galois group when is odd.

So what about when is even? By considering the fixed field of a Sylow-2-subgroup, we see that if appears as a Galois group, then it is the Galois closure of a degree 8 extension.

Now the degree 8 extensions of are finite and number and have all been computed, together with their Galois groups. They can be found at this online database. A quick examination shows that our binary tetrahedral group does not appear.

But it does appear as an inertia group. So is not a Galois group over , but is a Galois group over the unramified quadratic extension of .