Witt rings under odd degree extensions
Let \(K\) be a separable odd degree extension of the field \(F\). Then the Witt ring \(WK\) can be viewed as a \(WF\)-module and, on the other hand, the kernel \(\ker s_ *\) of the Scharlau transfer \(s_ *: WK\to WF\) is also a \(WF\)-module. The main theme of the paper is to decide whether or not these are projective or free \(WF\)-modules. Since \(WF\) need not be Noetherian and \(\ker s_ *\) need not be finitely generated, a generalized theory of attached primes is used to answer the main question. It turns out that \(WK\) is not a projective \(WF\)-module if some ordering on \(F\) extends uniquely to \(K\). An example is produced to show that \(WK\) may be a free \(WF\)-module if \(F\) is real and no ordering extends uniquely. When \(F\) is nonreal and nonrigid, then \(WK\) is not a projective \(WF\)-module provided \(K/F\) is Galois and \(WK\) is Noetherian.
scientific article; zbMATH DE number 509099