Let FF be a number field with a fixed embedding FhookrightarrowmathbbCFhookrightarrowmathbbC such that the restriction of complex conjugation from mathbbCmathbbC to FF is in Gal(F/mathbbQ)(F/mathbbQ) and fix a Hermitian inner product langlev,wrangle=overlinev1w1+cdots+overlinevnwnlanglev,wrangle=overlinev1w1+cdots+overlinevnwn on mathbbCnmathbbCn (with respect to the standard basis of mathbbCnmathbbCn. In particular, this restricts to a Hermitian inner product on FnFn.
Now suppose we are given a unitary matrix UU on FnFn with respect to that inner product. It is well known (independent of unitarity) that UU is diagonalizable over some extension E/FE/F - for instance, take EE to be a splitting field of the minimal polynomial of UU. This means there is a matrix MM over EE such that MUM−1MUM−1 is diagonal in the standard basis.
What if we instead want a unitary WW such that WUWdaggerWUWdagger is diagonal? This can be accomplished by working over a bigger extension E′/E that includes some extra square roots of elements of E. Namely, given any M that diagonalizes U over E, just add in the square roots of the eigenvalues of MdaggerM.
Now for my question:
Is there any sort of intrisic (i.e. independent of a choice of M) understanding of the extension E′? By understanding, I mean things like: is there a nice way of describing its generators over E? Can anything be said about its Galois group in general? When is it a semidirect product?
My actual interest is in the case where F is cyclotomic and U has finite order (and thus has roots of unity as eigenvalues, so E is another cyclotomic field). Any advice on what is known in this specific, or otherwise the general case, would be be much appreciated.
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