Singular elements and the Witt equivalence of rings of algebraic integers
Singular elements of a ring of \(S\)-integers of a global field are meant to be the non-zero elements of the field having even \(\mathfrak{p}\)-orders at all primes \(\mathfrak{p}\) not in \(S\). The authors study the group of singular elements in quadratic extensions of global fields, and in the case when the class number of the ring of \(S\)-integers is odd, they find a basis for the group (modulo squares). This is then used to discuss the existence of isomorphisms between the Witt rings of \(S\)- and \(S'\)-integers in two global fields \(K\) and \(K'\). The most complete results are obtained in the case of biquadratic extensions of nonreal number fields and of global function fields. In these cases necessary and sufficient conditions are found for the existence of strong isomorphisms between Witt rings of \(S\)- and \(S'\)-integers.
scientific article; zbMATH DE number 5565210