Inductive rings and fields

Ordered Rings and Fields

Witt rings of quadratically presentable fields

This paper introduces an approach to the axiomatic theory of quadratic forms based on {tmem{presentable}} partially ordered sets, that is partially ordered sets subject to additional conditions which amount to a strong form of local presentability. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of {tmem{quadratically p...

Supersimple fields and division rings

It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.

Commutative Rings and Finite Fields

Complete fields and valuation rings

In order to make further progress in our investigation of finite extensions L/K of the fraction field K of a Dedekind domain A, and in particular, to determine the primes p of K that ramify in L, we introduce a new tool that allows us to “localize” fields. We have seen how useful it can be to localize the ring A at a prime ideal p: this yields a discrete valuation ring Ap, a principal ideal dom...