10. Indecomposable Division Algebras

A central division algebra D over a field F is said to be decomposable if D=D ₁⊗ _F D ₂ for some proper subalgebras D ₁, D ₂ of D; otherwise, it is indecomposable. In this chapter we give examples of indecomposable algebras, emphasizing constructions that use valuation theory. In light of the primary decomposition, we restrict attention to division algebras of prime power degree. Indecomposable algebras of exponent p ² or higher are relatively easy to construct as “p-th roots” of other division algebras. Such constructions are discussed in §10.1, where we also give an example of an indecomposable division algebra D that becomes decomposable after a scalar extension of degree prime to \(\operatorname {\mathit{deg}}D\). §10.2 focuses on the more difficult case of indecomposables of prime exponent. We give in  §10.2.1 a criterion of Jacob to test the decomposability of a tame semiramified division algebra of prime exponent over a Henselian field. This criterion yields examples of exponent 2 and degree 8 in  §10.2.2, and of exponent p≠2 and degree p ^r with r≥2 in §10.2.4. The last section, §10.3, deals with complete decompositions into tensor products of symbol algebras. The main result is Th. 10.26, which relates armatures in an inertially split division algebra over a Henselian field to armatures in special representatives of its specialization coset.