See "Finding the distance between two circles in three-dimensional space," by C.A. Neff,
in the *IBM Journal of Research and Development*, Volume 34, Number 5, Page 770 (1990).
[IBM link][1]. From the Abstract:

> We show, by combining a theorem about solvable permutation groups and some explicit calculations with a computer algebra system, that, in general, the distance between two circles is an algebraic function of the parameters defining them, but that this function is not solvable in terms of radicals.


  [1]: http://domino.research.ibm.com/tchjr/journalindex.nsf/9fe6a820aae67ad785256547004d8af0/f6bc73813224c24e85256bfa0067fc5e!OpenDocument