The authors develop a general valuation theory for finite dimensional division algebras over Henselian fields. More strictly, let k be a valued field and \({\mathcal D}\) be a finite dimensional central division algebra over it, \(v_{{\mathcal D}}^ a \)valuation on \({\mathcal D}\), \(\Gamma_{{\mathcal D}}\) the value group, V the valuation ring of \(v_{{\mathcal D}}\) and \(M_{{\mathcal D}}\) its maximal ideal, \(\bar {\mathcal D}=V_{{\mathcal D}}/M_{{\mathcal D}}\). The importance of the finite group \(\Gamma_{{\mathcal D}}/\Gamma_ k\) and the center \(Z(\bar {\mathcal D})\) of \(\bar {\mathcal D}\) (for any ring A Z(A) denotes the center of A) follows from the fundamental homomorphism \(\theta_{{\mathcal D}}: \Gamma_{{\mathcal D}}/\Gamma_{Z({\mathcal D})}\to G(Z(\bar D)/\overline{Z({\mathcal D})})\), where G(Z(\(\bar {\mathcal D})/\overline{Z({\mathcal D})})\) is the Galois group of the extension Z(\(\bar {\mathcal D})/\overline{Z({\mathcal D})}\) and \(\theta_{{\mathcal D}}\) is induced by conjugation by elements of \({\mathcal D}^*\). The authors prove that \(\theta_{{\mathcal D}}\) is surjective and Z(\(\bar {\mathcal D})\) is the compositum of an abelian Galois extension and a purely inseparable extension of \(\overline{Z({\mathcal D})}.\)
Suppose that for the rest of the review k is a Henselian field. The authors give a description of the subgroup of the Brauer group Br k of k which is represented by central inertial division algebras over k (one says that a division algebra \({\mathcal D}\) over k is inertial if [\({\mathcal D}:k]=[\bar {\mathcal D}:\bar k]\) and Z(\(\bar {\mathcal D})\) is separable over \(\bar k\)). Furthermore they prove that every division algebra \({\mathcal D}\) finite dimensional over k with Z(\(\bar {\mathcal D})\) separable over \(\bar k\) has an inertial subalgebra, i.e. a subalgebra B such that \(\bar B=\bar {\mathcal D}\) and \([B:k]=[\bar B:\bar k]\). In the paper there are some results about the division algebra \({\mathcal D}_ F\) which is similar to the algebra \({\mathcal D}\otimes_ kF\) where F is any inertial field extension of k. Then the authors consider the class of so-called nicely semiramified division algebras, (i.e., subalgebras which have a maximal subfield inertial over k and another maximal subfield totally ramified over k). It is proved that each such algebra is a tensor product of nicely semiramified algebras.
The next object of consideration is inertially split algebras, i.e. algebras which have a maximal subfield inertial over k. It is proved in the paper that every inertially split division algebra \({\mathcal D}\) is similar to \(B\otimes_ kN\) where B is inertial over k and N is nicely semiramified. After these preliminary considerations the authors attack the problem of investigation of tame division algebras. They show that every such algebra is similar to \(S\otimes_ kT\), where S is inertially split and T is totally ramified over k. Furthermore they prove that \(\Gamma_{{\mathcal D}}=\Gamma_ S+\Gamma_ T\) and show how Z(\(\bar {\mathcal D})\) and \(\theta_{{\mathcal D}}\) are connected with \(Z(\bar S)\), \(\theta_ s\) and \(\Gamma_ S\cap \Gamma_ T\). At the end of the paper some interesting examples are given.
Remark. Some results of the paper were obtained independently by V. P. Platonov and the reviewer [in their papers Dokl. Akad. Nauk SSSR 297, 294-298 (1987); English transl. in Sov. Math., Dokl. 36, 468-472 (1988; Zbl 0667.16018) and ibid. 297, 542-547 (1987); English transl. in ibid. 36, 502-506 (1988; Zbl 0669.16016)]. In these papers there are also other results about Henselian division algebras including a complete classification of Henselian tame division algebras.