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Real number field


I have matrices of the following form: M = [[ sqrt(2), 1], [14, cos(pi/8)]].

All of the entries are totally real algebraic numbers. I would like to study the signature of the conjugates of M.

So basically, I would like to create a number field containing the two elements sqrt(2), cos(pi/8) and use the different embeddings into R (or C) to study the signature.

The following does not work: QQ[sqrt(2)][cos(pi/8)] because the minimal polynomial of cos(pi/8) is not irreducible over Q[sqrt(2)]. How can I get it to work?

I could compute it manually, but I would like to make the process automatic since the matrices are all of the above form (with expressions like sqrt(2), cos(pi/8) regardless of the fact that one can be written in terms of the other).

Thank you for your help.