Class number parity and unit signature
A number field is said to be \(2^ +\)-regular if it is totally real, contains exactly one dyadic prime, has odd \(S\)-class number and contains \(S\)-units of independent signs where \(S\) consists of the dyadic prime and all infinite primes of the field. The objective of this article is to classify quadratic extensions of \(2^ +\)-regular number fields according to parity of their class number and whether or not they contain units of independent signs. This leads to four types of fields depending on which combination of the two properties is satisfied. Families of \(2^ +\)- regular quadratic extensions of a \(2^ +\)-regular number field \(F\) are defined by means of the square class in the dyadic completion of \(F\). All fields belonging to a family are shown to be of the same type. Moreover, there are \(2^{r+1}\) infinite families of \(2^ +\)-regular extensions of \(F\), where \(r=[F:Q]\). Finally, depending on the type of \(F\), the number of families of quadratic extensions of \(F\) of each type is determined.
scientific article; zbMATH DE number 108929