Post Made Community Wiki by Anton Geraschenko♦♦
1

I can't answer in general, but I will speak to some examples. For the problem on finite unions of subspaces, I typed finite union subspaces vector into Google, and the top of the resulting list was

On the representation of vector spaces as a finite union of subspaces by J Luh - 1972 FINITE UNION OF SUBSPACES. By. J. LUH (Raleigh). It has been proved in [1] that if a vector space V over a field F is a union of n (finite) proper subspaces ... www.springerlink.com/index/GN711127680W310R.pdf

For the ${\bf Q}(\sqrt2,\sqrt7)$ problem, I typed in class number biquadratic and got a number of results that look like they might be helpful. Typing in class number computation biquadratic, the top result was

A Computation of Some Bi-Quadratic Class Numbers by H Cohn - 1958 2 99191. A Computation of Some Bi-Quadratic Class Numbers. By Harvey Cohn. A fascinating chapter in computational number theory began when Lagrange ... www.jstor.org/stable/2002024

which looks like it might be useful, as do some of the other returns.