Unit Signatures in Real Biquadratic and Multiquadratic Number Fields

Abstract

We consider the signature rank of the units in real multiquadratic fields. When the three quadratic subfields of a real biquadratic field $K$ either (a) all have signature rank 2 (that is, fundamental units of norm $-1$), or (b) all have signature rank 1 (that is, have totally positive fundamental units), we provide explicit examples to show there exist infinitely many $K$ having each of the possible unit signature ranks (namely signature rank 3 or 4 in case (a) and signature rank 1,2, or 3 in case (b)). We make some additional remarks for higher rank real multiquadratic fields, in particular proving the rank of the totally positive units modulo squares in such extensions (hence also in the totally real subfield of cyclotomic fields) can be arbitrarily large.