A Note on the Equivalence of the Parity of Class Numbers and the Signature Ranks of Units in Cyclotomic Fields

Abstract

We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal.2010 Mathematics Subject Classification. 11R18 (primary), and 11R27, 11R29 (secondary).

Signature ranks of units in cyclotomic extensions of abelian number fields

“…We also note that the deficiency of the circular units is at least the deficiency for the full group of units, but may be strictly larger: for the field Q(ζ 163 ) + the circular unit deficiency is 2, while the deficiency for the full group of units is 0 (see [3]).…”

“…A nice proof of this, using the fact that over F 2 the only irreducible representation of a 2-group is the trivial representation, can be found in [7]. Another nice proof of Weber's result can be found in [3].…”

Unit Signatures in Real Biquadratic and Multiquadratic Number Fields

“…In fact the strict class number of the fields K in Theorem 1 is odd (so equal to the class number), as follows. The extension K/Q( √ p 1 ) is of degree 2, unramified outside the prime (p 2 ), and totally ramified at this prime, so its strict class number is odd if and only if the strict class number of Q( √ p 1 ) is odd (see [D1,Lemma]), and Q( √ p 1 ) has odd class number by genus theory. Remark 6.…”