Algebraic independence of elements in immediate extensions of valued fields

Abstract

Refining a constructive combinatorial method due to MacLane and Schilling, we give several criteria for a valued field that guarantee that all of its maximal immediate extensions have infinite transcendence degree. If the value group of the field has countable cofinality, then these criteria give the same information for the completions of the field. The criteria have applications to the classification of valuations on rational function fields. We also apply the criteria to the question which extensions of a maximal valued field, algebraic or of finite transcendence degree, are again maximal. In the case of valued fields of infinite $p$-degree, we obtain the worst possible examples of nonuniqueness of maximal immediate extensions: fields which admit an algebraic maximal immediate extension as well as one of infinite transcendence degree.