Witt groups of algebraic integers
Let \(F\) be an algebraic number field and let \(D\) be its ring of integers. There is an injective group homomorphism \(WD\to WF\), where \(WD\) and \(WF\) are Witt rings of \(D\) and \(F\), respectively. The author is interested in determining the structure of the additive group \(WD\) and in deciding when \(WD\to WF\) splits. He determines the structure of the abelian group \(WD\) in terms of arithmetic invariants of the field \(F\). He also has some necessary and sufficient conditions for the splitting problem. For instance, if \(F\) is totally imaginary, then the condition is that every element of \(WD\) 2-divisible in \(WF\) be 2-divisible in \(WD\). For real quadratic number fields the condition is given in terms of prime factors of the discriminant, and for imaginary quadratic fields it is shown that \(WD\to WF\) always splits. \{In case of quadratic number fields there is an overlap with two papers written at the University of Bielefeld by \textit{W. Colberg} [Wittringe von Dedekindringen, Diplomarbeit, 1975] and \textit{R. Münstermann} [Der Wittring des Ringes der ganzen Zahlen eines quadratischen Zahlkörpers, Staatsexamensarbeit, 1983]. Colberg determines the additive structure of \(WD\) for any quadratic number field \(F\), and Münstermann, in addition to determining the generators of direct summands in a decomposition of \(WD\) for a quadratic field \(F\), shows how to multiply the generators thus describing completely the ring structure of \(WD\).\}
scientific article; zbMATH DE number 4061323