Small extensions of Witt rings (Q1305103)

The author investigates the (potential) behaviour of Witt rings of odd degree field extension. An abstract Witt ring (in the sense of Marshall) is of elementary type if it can be built up in finitely many steps form \({\mathbb Z}\), \({\mathbb Z}/2{\mathbb Z}\), \({\mathbb Z}/4{\mathbb Z}\) and Witt rings of local type using the operations of direct product and group ring formation. Let \(R\) be an abstract Witt ring with associated group of one dimensional forms \(G(R)\) and let \(H\) denote a subgroup of \(G(R)\). An \(H\)-extension \(S\) of \(R\) is defined as follows: \(G(S)=\{1,\alpha\}G(R)\) for a suitable element \(\alpha \in G(S)\) and for all \(x\in G(R)\) define: \(D_S\langle 1,-x\rangle=D_R\langle 1,-x\rangle\) if \(x\not \in H\), \(D_S\langle 1,-x\rangle=\{1,\alpha\}D_R\langle 1,-x\rangle\) if \(x\in H\), \(D_S\langle 1,-\alpha x\rangle=\{1,-\alpha x\}(D_R\langle 1,-x\rangle \cap H)\). In [\textit{R. Fitzgerald}, Pac. J. Math. 158, 121-143 (1993; Zbl 0790.11034)] the author proved that if \(K/F\) is an odd degree field extension with \([G(K):G(F)]=2\) then \(W(K)\) is an \(H\)-extension of \(W(F)\). In the paper under review the author proves that if \(R\) is of elementary type then \(S\) is a Witt ring of elementary type. It is worth noticing that no example of field extension yielding an \(H\)-extension is known, but every \(H\)-extension \(W(F)\) is realized by a field, provided \(W(F)\) is of elementary type.