TITLE: Cyclotomic polynomials explicitly solvable??
QUESTION [10 upvotes]: I don't know why I'm having trouble with this, but I can't quite see whether the cyclotomic polynomials are considered solvable. Obvioulsy we can write the solution of the nth cyclotomic polynomial as the nth root of unity, which seems to be a perfectly good algebraic solution; but aren't there usually "better" solutions? I've done the fifth and seventh, and I think Gauss is credited for solving the eleventh if I'm not mistaken; the seventeenth falls out of the construction of the regular 17-gon...but I'm still not clear: are all of them, in theory, solvable "explicitly", which is to say beyond the level of just saying that the nth root of unity is a solution?

REPLY [12 votes]: The method of Lagrange resolvents succeeds to solve cyclotomic equations in radicals (in the useful sense, not the "nth root of 1" sense). This is discussed in fairly elementary terms, with examples, in chapter 19 of my algebra book/course-notes, at http://www.math.umn.edu/~garrett/m/algebra/
However, that essentially-elementary approach becomes burdensome quickly, certainly difficult to do or understand by hand calculation. But if we observe that the resolvents are _Gauss_sums_, then admixture of some ideas of Kummer and Eisenstein gives a much-improved, but less elementary approach to expression of roots of unity by radicals. This is discussed, with several examples, in http://www.math.umn.edu/~garrett/m/v/kummer_eis.pdf