The classification of the structure of real and complex Clifford algebras is well-known. Most sources concentrate only upon that case. Another resource. Also, I'm only interested in nondegenerate forms, at this point, and I'd prefer to skip characteristic $2$ if it complicates the answer.
The interesting thing is the periodicity exhibited in their structures as the dimension and signature changes.
But the theory of bilinear forms is well-developed over all fields, so I am very curious what happens outside of the idyllic cases of $\mathbb R$ and $\mathbb C$. I believe I've seen somewhere how to prove how Clifford algebras over nondegenerate spaces are all semisimple rings (at least, outside of characteristic $2$.)
Correct me if I'm wrong, but if I remember right, the only thing you need to reduce quadratic forms to a "signature" is to either use an ordered which contains the square roots of all its positive members, or to use a field that's quadratically closed altogether.
Is the structure/periodicity of Clifford algebras of algebras over fields described in the last paragraph known?
Alternatively, we could stop caring about general bilinear forms and just ask for a classification of those arising from a matrix with $1$'s and $-1$'s on the diagonal.
Is the structure/periodicity of Clifford algebras for these particular forms over fields known?
I haven't had a lot of luck with the literature. I could believe it if everything had to exhibit periodicity $1, 2, 4,$ or $8$, because that seems to be the way geometry likes to roll.