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Glasgow Mathematical Journal (1)

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INFINITE HILBERT CLASS FIELD TOWERS OVER CYCLOTOMIC FIELDS
IGOR E. SHPARLINSKI
Journal:

Glasgow Mathematical Journal / Volume 50 / Issue 1 / January 2008

Published online by Cambridge University Press:

01 January 2008, pp. 27-32

Print publication:

January 2008
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We use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field 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 for some p≥m0.3385 + o(1). These results have immediate applications to the divisibility properties of the class number of .