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Splitting of prime ideals and valuations
Abstract
In this thesis, we investigate the splitting of rank-one valuations in field extensions and related questions on the splitting of prime ideals of affine domains in extensions of the quotient field. These properties have many applications in Galois theory and to the study of multiplicative groups of fields. In chapter 1 we outline the thesis and introduce notation and some basic results, which will be used in the following chapters. In chapter 2, we generalize the well-known fact that for every finite algebraic extension L/K of algebraic number fields there are infinitely many discrete rank-one valuations of K that split completely in L. We show that for every finite separable extension L/K there are card(K) inequivalent rank-one valuations of K that split completely in L, unless K is algebraic over a finite field. In many cases one can choose the valuations to be discrete. In chapter 3, we deal with the factorization of prime ideals of affine domains under finite separable extensions. The main result runs as follows: Let D be a domain of finite type over the ring of integers or over a field. Let K be the quotient field of D, let L/K be a finite separable extension, and let B be the integral closure of D in L. Then, for every i with 1 $\leq$ i $\leq$ dim(D), there are infinitely many prime ideals P of height i in D such that PB$\sb{\rm P}$ is a product of exactly (L: K) distinct prime ideals of B$\sb{\rm P}$ and the residue fields are naturally isomorphic. Hence there are infinitely many maximal ideals m of B such that m + D = B. In the last chapter of this thesis we apply our main results to the study of the norm function, obtaining an analogue of Hilbert's Theorem 90 that applies to any finite Galois extension, not necessarily cyclic. We also obtain applications to the theory of rings satisfying a polynomial identity.
Subject Area
Mathematics
Recommended Citation
Jia, Bao-Ping, "Splitting of prime ideals and valuations" (1990). ETD collection for University of Nebraska-Lincoln. AAI9118456.
https://digitalcommons.unl.edu/dissertations/AAI9118456