Quaternion algebras with derivations (Q1979278)

The goal of this paper is to study various questions regarding central simple algebras with derivations, with an emphasis on quaternion algebras. \textit{S. A. Amitsur} [Commun. Algebra 10, 797--803 (1982; Zbl 0486.16014)] proved that for every maximal subfield \(L\) of a central simple \(k\)-algebra \(A\), there is a derivation on \(A\) mapping \(L\) to \(L\). For example, when \(\operatorname{char}(k)=2\) and \(A\) is a quaternion algebra, the latter is generated over \(k\) by some \(x\) and \(y\) satisfying \(x^2+x=\alpha\), \(y^2=\beta\) and \(xy+yx=y\) where \(\alpha \in k\) and \(\beta\in k^\times\). In this case, the algebra \(A\) is endowed with a derivation \(\delta\) mapping any \(v\) to \(xv+vx\), and thus \(\delta\) stabilizes the quadratic subfield \(k(y)\) of \(A\). This raises the question of whether every derivation must stabilize a maximal subfield. This was answered in the negative in the paper under discussion (Proposition 4.6), where it is proved that every quaternion algebra over \(\mathbb{Q}\) admits a derivation stabilizing no quadratic subfield. Another theme in this paper is the study of splitting fields of central simple algebras with derivations, a notion introduced in [\textit{L. Juan} and \textit{A. R. Magid}, Proc. Am. Math. Soc. 136, No. 6, 1911--1918 (2008; Zbl 1223.12006)]. In Theorem 4.1 of the paper under discussion, the authors provide explicit constructions of differential fields that split differential quaternion algebras. They prove that there are (differential) fields of transcendence degree 1 over \(\mathbb{Q}\), splitting a differential quaternion \(\mathbb{Q}\)-algebra. They also prove that no algebraic extension splits a differential quaternion division \(\mathbb{Q}\)-algebra (Theorem 4.5). They conclude the paper by initiating a study of differential quaternion algebras over function fields. In Proposition 5.1, they show that those quaternion algebras over a function field \(k(t)\), which are not defined over \(k\), are of the form \((\alpha, \delta(\alpha))_{k(t)}\), where \(\alpha \in k(t) \setminus k\) and \(\delta\) is a derivation on \(k\). Then, in Theorem 5.2, given \(\alpha \in k(t)\) they construct suitable derivations \(\delta\) on \(k(t)\) so that \((\alpha, \delta(\alpha))_{k(t)}\) is a division algebra.