Primary ideals in Witt rings
For a field F, let WF denote the Witt ring of classes of non-degenerate symmetric bilinear forms. Spec WF was described, ca. 1970, by \textit{D. K. Harrison} (University of Kentucky Notes) and independently by \textit{F. Lorenz} and \textit{C. Leicht} [Invent. Math. 10, 82-88 (1970; Zbl 0227.13015)]: IF, the ideal of classes of even dimensional forms is always in Spec WF, if F is formally real, \(\alpha\) is an ordering of F, and p is a rational prime, then all other elements of Spec WF are the ideals \({\mathcal P}(\alpha)\), \({\mathcal P}(\alpha,p)\) of classes of forms whose Sylvester signature for \(\alpha\) is 0, or divisible by p, respectively. In this paper, the primary ideals of WF are calculated. These are, in addition to the elements of Spec WF, any ideal containing \(2^ k\) for \(k\in {\mathbb{N}}\), and, if F is formally real, the ideals \({\mathcal P}(\alpha,p^ n)\) of classes of forms whose Sylvester signatures for \(\alpha\) are divisible by \(p^ n.\) The height of F, h(F), is the smallest power of 2, if one exists, such that h(F) Rad WF\(=0\), or infinity otherwise. It is shown that, if F is formally real, then every ideal of WF has a primary decomposition if and only if, the number of orderings of F and h(F) are finite. Several consequences of this are found, e.g., if F satisfies the two hypotheses and an ideal I of WF contains the class of an odd dimensional form, then \(I\supset Rad WF\). Further for these fields, it is shown, i.a., that \({\mathcal P}(\alpha,3)\) principal for all \(\alpha\), is equivalent to unique factorization into irreducible elements for the classes of odd dimensional forms, and that F is SAP iff every ideal of WF containing a class of an odd dimensional form, is principal with generator a class of a non-totally indefinite form. The last topic is a determination of necessary and sufficient conditions on an ideal I in WF, where F is formally real with both finite height and only finitely many orderings, to satisfy \(I=ann(ann(I))\).
scientific article; zbMATH DE number 3916372