A Note on the Linkage of Hurwitz Algebras

Triple linkage of quadratic Pfister forms

“…A maximal subfield K of a quaternion algebra over F is a quadratic field extension of F. When char(F) = 2, K/F can be either separable or inseparable, and one can refine the definition of linkage accordingly: a set of quaternion algebras is separably (inseparably) linked if they share a common separable (inseparable) quadratic field extension of F. It was observed in [Dra75] that inseparable linkage for pairs of quaternion algebras implies separable linkage, and a counter example for the converse was provided in [Lam02]. This observation was extended to pairs of Hurwitz algebras in [EV05] and pairs of cyclic p-algebras of any prime degree in [Cha15]. We extend the notion of separable and inseparable linkage of quaternion algebras to arbitrary n-fold Pfister forms in the following way: a set of quadratic n-fold Pfister forms are separably (inseparably) linked if there exists a quadratic (bilinear) (n−1)-fold Pfister form which is a common factor to all the forms in the set.…”

Linkage of sets of Quaternion Algebras in characteristic 2

“…It was proven in [Cha15] that if A 1 and A 2 are inseparably linked then they are also cyclically linked and counterexamples were provided for the converse statement. The special case of quaternion algebras had been proven earlier in [Dra83], [Lam02] and [EV05]. Note that quadratic field extensions are either cyclic or purely inseparable, and therefore if quaternion algebras A 1 , .…”

Total linkage of quaternion algebras and Pfister forms in characteristic two

“…One can show that if two quaternion algebras share an inseparable quadratic subfield then they share a separable quadratic subfield, but that the converse is not always true (see [15]. This result was also generalized to Hurwitz algebras in [4]). This motivated the consideration of two different types of 'linkage' in characteristic 2, depending on whether the shared subfield of two quaternion algebras over F was a quadratic separable or a quadratic inseparable extension of F.…”