The Witt ring kernel for a fourth degree field extension and related problems
Let \(F\) be a field of characteristic not \(2\). An important question in the algebraic theory of quadratic forms is the determination of the kernel \(W(L/F)\) of the restriction map \(W(F)\to W(L)\) from the Witt ring of the field \(F\) to that of an extension field \(L\) over \(F\). It is known that \(W(L/F)=0\) for odd degree extensions due to the so-called Springer's theorem, and for \(L=F(\sqrt{d})\) the kernel is generated by the class of the norm form \(\langle 1,-d\rangle\) of the quadratic extension. In the present paper, the author determines \(W(L/F)\) in the case \([L:F]=4\) by explicitly describing the generators of this ideal. Previously, the kernel of such degree \(4\) extensions has only been determined in the situation where \(L\) contains a quadratic subextension of \(F\). It turns out that in general the kernel is generated by forms of type \(\langle 1,-d\rangle\) with \(F\subset F(\sqrt{d})\subset L\), and by \(2\)-fold Pfister forms \(\langle\!\langle p(\alpha),-\alpha\rangle\!\rangle\), where \(p(X)\in F[X]\) is the resolvent cubic of the minimal polynomial \(X^4+aX^2+bX+c\in F[X]\) of a suitable primitive element generating the extension \(L/F\). It is furthermore shown (invoking the Milnor conjecture proved by Voevodsky) that for such degree \(4\) extensions \(L/F\), one has \(W(L/F)\cap I^n(F)=(W(L/F)\cap I^2(F))I^{n-2}F\) for \(n\geq 2\), where \(I^n(F)\) denotes the \(n\)-th power of the fundamental ideal of classes of even-dimensional forms in \(W(F)\). The author also provides further applications of his results. He gives a necessary and sufficient criterion for when the tensor product \(Q\otimes_F B\) of a quaternion algebra \(Q\) and a biquaternion algebra \(B\) over \(F\) is not a division algebra. In the last section of the paper, the author studies degree \(4\) extensions \(L/F\) containing a quadratic subextension. The smallest Galois extension \(M\) of \(F\) containing such an \(L\) is either \(L\) itself, or (if \(L/F\) is not Galois) a quadratic extension of \(L\). The author constructs several exact sequences involving the Witt rings of various subextensions of \(M\) and shows that if \(F\) is not formally real, then \(W(M/F)\) is generated by the kernels \(W(K/F)\) of subextensions \(F\subseteq K\subseteq M\) with \([K:F]=4\).
scientific article; zbMATH DE number 5624920
scientific article