4. Existence and Fundamental Properties of Gauges

This chapter is crucial for the study of gauges. Let F be a field with valuation v, and let (F _h ,v _h ) be the Henselization of (F,v). In §4.1 we prove Morandi’s criterion, Th. 4.1, that v extends to a division algebra D with center F if and only if D⊗ _F F _h is a division algebra. Among the applications are a primary decomposition theorem and an Ostrowski-type defect theorem, Th. 4.3, for valued division algebras. The valuation v is said to be defectless in D if equality holds in the Fundamental Inequality for the extension of v to D. In  §4.2 we extend the notion of defectlessness of v to any semisimple F-algebra  A by considering the simple components of A⊗ _F F _h . In §4.3 we prove key classification results for gauges. We show in Th. 4.26 that if S is a central simple F _h -algebra, then any v _h -gauge on S is an \(\operatorname {\mathit{End}}\)-gauge as in  Prop. 3.​34. We also show how v-gauges on a semisimple F-algebra A are built from gauges on the simple components A _i of A for the extensions of v to  Z(A _i ). §4.4 is devoted to proving Th. 4.50, which says that a semisimple F-algebra has a v-gauge if and only if v is defectless in A.