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Let φ and ψ be regular quadratic forms over a field F of characteristic different from 2. We say that ψ is a quasisubform of φ if there is a∈F* such that aψ is a subform of φ. Let L/F be an odd deg...

For a given field F of characteristic different from 2 and $$a,b,d\in F^*$$ we construct an invariant $$\mathrm{inv}$$ for an element $$D\in \,_2\mathrm{Br}(F(\sqrt{a},\sqrt{b},\sqrt{d})/F)$$ . This invariant takes value in the quotient group $$\begin{aligned} H^3(F,\mu _2)/D\cup {\mathrm{N}_{\mathrm{F}\left( \sqrt{\mathrm{d}}, \sqrt{\mathrm{ab}}\right) /\mathrm{F}}}F\left( \sqrt{d},\sqrt{ab}\right) ^*. \end{aligned}$$ Let k be a field, let $$k(\sqrt{a},\sqrt{b},\sqrt{d})/k$$ be a triquadratic field extension. We apply the invariant $$\mathrm{inv}$$ and a few deep results from algebraic geometry and K-theory to construct a field extension K/k with $$\mathrm{cd}_2 K=3$$ , and an indecomposable cross product algebra of exponent 2 with respect to the extension $$K(\sqrt{a},\sqrt{b},\sqrt{d})/K$$ . Using the invariant $$\mathrm{inv}$$ , we also prove the following odd degree descent statement: Assume $$D\in \,_2\mathrm{Br}(F)$$ , $$b,d\in F^*$$ , L/F is an odd degree extension. Assume also that $$D_{L(\sqrt{b},\sqrt{d})}=Q_{L(\sqrt{b},\sqrt{d})}$$ , where Q is a quaternion algebra defined over L. Then there exists a quaternion algebra $$\widetilde{Q}$$ defined over F such that $$D_{F(\sqrt{b},\sqrt{d})}=\widetilde{Q}_{F(\sqrt{b},\sqrt{d})}$$ . As a consequence we get that if $$\phi \in I^2(F)$$ is a form such that $${(\phi _{L(\sqrt{b},\sqrt{d})})}_{an}$$ is defined over L, and $$\dim {(\phi _{L(\sqrt{b},\sqrt{d})})}_{an} =4$$ , then $${(\phi _{F(\sqrt{b},\sqrt{d})})}_{an}$$ is defined over F.

Let F be a field, charF=2,−1∈F*. Let further D1, D2 be biquaternion algebras over F with Albert forms φ1,φ2. We prove that the dimension of anisotropic part of the form φ1⊗φ2 is at most 24, provide...

The homology groups h1(l/k), h2(l/k), and h3(l/k) of the Brauer complex for a triquadratic field extension l=k(a,b,c) are studied. In particular, given D∈APTARAMOPREFIX2 Br (k(a,b,c)/k), we find equivalent conditions for the image of D in h2(l/k) to be zero. We consider as well the second divided power operation γ2:APTARAMOPREFIX2 Br (l/k)→H4(k,Z/2Z), and show that there are nonstandard elements with respect to γ2. Further, a natural transformation h2⊗h1→H3, which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension F/k such that the cohomology group h1(F(a,b,c)/F) of the Brauer complex contains the images of prescribed elements of k∗, provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions L/F with nontrivial h3(L/F) are given. As a consequence we show that the homology group h3(L/F) can be arbitrarily big.

Let F be a field of characteristic different from 2, and let Q1; Q2; Q3 be quaternion algebras over F such that any element in Br.F/ generated by Q1; Q2; Q3 has index at most 2. For the triplef Q1; Q2; Q3g we construct a certain invariant, lying in I 3 .F/, which is a 4-fold Pfister form, provided the algebras Q1; Q2; Q3 have a common slot. Among other results we prove that the algebras Q1; Q2; Q3 have a common slot if and only if the torsion of the group CH 2 .X1 X2 X3/ is zero, where X i is the projective conic associated with the algebra Q i .

Let A be a central simple algebra over the field of rational functions in one variable over an arbitrary field of characteristic different from 2. If the Schur index of A is not divisible by the characteristic and its ramification locus has degree at most 3, then A is Brauer-equivalent to the tensor product of a quaternion algebra and a constant central division algebra D. The index of A is computed in terms of D and the ramification of A. The result is used to construct various examples of division algebras over rational function fields.