DOUBLE COVERINGS AND UNIT SQUARE PROBLEMS FOR CYCLOTOMIC FIELDS

Abstract

In this paper, using the theory of double coverings of cyclotomic fields, we give a formula for [Formula: see text], where K = ℚ(ζn), G = Gal (K/ℚ), 𝔽2 = ℤ/2ℤ and UK is the unit group of K. We explicitly determine all the cyclotomic fields K = ℚ(ζn) such that [Formula: see text]. Then we apply it to the unit square problem raised in [Y. Li and X. Zhang, Global unit squares and local unit squares, J. Number Theory128 (2008) 2687–2694]. In particular, we prove that the unit square problem does not hold for ℚ(ζn) if n has more than three distinct prime factors, i.e. for each odd prime p, there exists a unit, which is a square in all local fields ℚ(ζn)v with v | p but not a square in ℚ(ζn), if n has more than three distinct prime factors.