1. Valuations on Division Rings

In this chapter we introduce the central object of study in this book: valuations on a division algebra D finite-dimensional over its center  F. In §1.1 we define valuations and describe the associated structures familiar from commutative valuation theory: the valuation ring \(\mathcal {O}_{D}\), its unique maximal left and maximal right ideal \(\mathfrak {m}_{D}\), the residue division algebra \(\overline{D}\), and the value group Γ _D . We also describe an important and distinctively noncommutative feature, namely a canonical homomorphism θ _D from  Γ _D to the automorphism group \(\operatorname {\mathit{Aut}}(Z(\overline{D})\big/\,\overline{F}\,)\); θ _D is induced by conjugation by elements of D ^×. In §1.2, after proving the “Fundamental Inequality for valued division algebras, we look at valuations on D from the perspective of F. We show that a valuation on F has at most one extension to D, and prove a criterion for when such an extension exists. When this occurs, we show that the field \(Z(\overline{D})\) is finite-dimensional and normal over \(\overline{F}\) and that θ _D is surjective. We also describe the technical adjustments needed to apply the classical method of “composition” of valuations to division algebras. The filtration on D induced by a valuation leads to an associated graded ring \(\operatorname {\mathsf {gr}}(D)\), which we describe in §1.3. Throughout the book we emphasize use of \(\operatorname {\mathsf {gr}}(D)\) to help understand the valuation on D. This chapter includes many examples of division algebras with valuations.