There are applications of the theory of cyclotomic fields to free actions of finite groups on $S^n$. The existence of such actions is tied to the class groups of certain cyclotomic fields $\mathbb{Q}(\mu_N)$ and their maximal real subfields $\mathbb{Q}(\mu_N)^+$.

You can find a brief introduction to this concept on p265 in Lang's Units and Class Groups in Number Theory and Algebraic Geometry: http://projecteuclid.org/euclid.bams/1183548780

Here are some of the relevant references he cites:

J. MILGRAM, Odd index subgroups of units in cyclotomic fields and applica- tions, Springer Lecture Notes no. 854 (1981).

J. MILGRAM, Patching techniques in surgery and the solution of the compact space form problem.

D. KUBERT, The 2-primary component of the ideal class group in cyclotomic fields.

C. T. C. WALL, Classification of hermitian forms VI, Ann. of Math. 103 (1976) pp. 1-80.