Nonexcellent finite field extensions of 2-primary degree
Let \(F\) be a field of characteristic different from \(2\). A field extension \(L/F\) is called excellent if for any quadratic form \(\varphi\) over \(F\), the anisotropic part of the form obtained by extending scalars to \(L\) is defined over \(F\). The extension \(L/F\) is called strongly excellent if for any extension \(K/F\) linearly disjoint with \(L/F\), the extension \(KL/F\) is excellent. For example, odd degree and quadratic extensions are well known to be strongly excellent. This paper considers the question as to whether the only strongly excellent extensions are those that can be realised as a quadratic extension followed by an odd degree extension. It is shown that any field extension \(L/F\) of degree \(2^m\) with \(m\geq 2\) is not strongly excellent. This is done via the following construction. First, it may be assumed that \(F\) has no finite odd degree extensions and that there exists an intermediate field \(F\subset k \subset L\) with \(k=F(\sqrt{u+v\sqrt{a}})\) and such that \(k/F\) is Galois. The proof is then divided into cases based on the Galois group \(G\) of \(k/F\), with the case \(G=\mathbb{Z}/4\mathbb{Z}\) being the most difficult. In this case, the fields \(F\) and \(L\) are replaced by \(F(x)\) and \(L(x)\) with \(x\) an indeterminate. A field extension \(K/F\) and a quaternion algebra \(D\) are then constructed such that \[ D\in {_2\mathrm{Br}}(K(\sqrt{x}, \sqrt{u+v\sqrt{a}})/K) \quad\mathrm{ and }\quad D\notin {_2\mathrm{Br}}(K(\sqrt{x})/K) + {_2\mathrm{Br}}(LK/K)\,. \] A quadratic form may be associated with this quaternion algebra, and the above conditions show that this quadratic form becomes isotropic over \(L\), but its anisotropic part over \(L\) is not defined over \(F\). The second of these two conditions on \(D\) is the more difficult, and requires computations with \(D\) in the cohomology groups \(H^1(K,\mathbb{Z}/2\mathbb{Z})\) and \(H^3(K,\mathbb{Z}/2\mathbb{Z})\). The results required for the construction of \(D\) are also applied to the following problem. Let \(L/F\) be a \(\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}\)-Galois extension, \(F_i\) (\(1\leq i\leq 3\)) be three intermediate quadratic extensions of \(F\) and \(N_i(F)=N_{F_i/F}F^\times_i\). It is shown that the quotient group \[ \frac{N_1(F)\cap N_2(F)\cap N_3(F)}{{F^\times}^2 N_{L/F}L^\times} \] can be arbitrarily large.
scientific article; zbMATH DE number 6666258
scientific article