INFINITE HILBERT CLASS FIELD TOWERS OVER CYCLOTOMIC FIELDS

Abstract

AbstractWe use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field $\Q(\exp(2 \pi i/m))$ has an infinite Hilbert p-class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log logm)1 + o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower over $\Q(\exp(2 \pi i/m))$ for some p≥m0.3385 + o(1). These results have immediate applications to the divisibility properties of the class number of $\Q(\exp(2 \pi i/m))$.