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~2 Q ! / Date an EXPOSITION on the KRONECKER-WEBER THEOREM An exposition on the Kronecker-Weber theorem Item Type Thesis Authors Baggett, Jason A. Download date 24/09/2021 09:05:39 Link to Item http://hdl.handle.net/11122/11349 AN EXPOSITION ON THE KRONECKER-WEBER THEOREM By Jason A. Baggett RECOMMENDED: Advisory Committee Chair Chair, Department of Mathematics APPROVED: Dean, College of Natural §sztfmce and Mathematics ; < r Dean of the Graduate School ~2 Q_! / Date AN EXPOSITION ON THE KRONECKER-WEBER THEOREM A THESIS Presented to the Faculty of the University of Alaska Fairbanks in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE By Jason A. Baggett, B.S. Fairbanks, Alaska May 2011 iii Abstract The Kronecker-Weber Theorem is a classification result from Algebraic Number Theory. Theorem (Kronecker-Weber). Every finite, abelian extension Qof is contained in a cyclo- tomic field. This result was originally proven by Leopold Kronecker in 1853. However, his proof had some gaps that were later filled by Heinrich Martin Weber in 1886 and David Hilbert in 1896. Hilbert's strategy for the proof eventually led to the creation of the field of mathematics called Class Field Theory, which is the study of finite, abelian extensions of arbitrary fields and is still an area of active research. Not only is the Kronecker-Weber Theorem surprising, its proof is truly amazing. The idea of the proof is that for a finite, Galois extension K of Q, there is a connection be­ tween the Galois group Gal(K/Q) and how primes of Z split in a certain subring R of K corresponding to Z in Q. When Gal(K/Q) is abelian, this connection is so stringent that the only possibility is that K is contained in a cyclotomic field. In this paper, we give an overview of field/Galois theory and what the Kronecker-Weber Theorem means. We also talk about the ring of integers R of K, how primes split in R, how splitting of primes is related to the Galois group Gal(K/Q), and finally give a proof of the Kronecker-Weber Theorem using these ideas. iv Table of Contents Page Signature Page ............................................................................................................... i Title Page ........................................................................................................................ ii Abstract ........................................................................................................................... iii Table of Contents ......................................................................................................... iv List of Figures ............................................................................................................... vi Acknowledgements ......................................................................................................... vii 1 Introduction 1 2 Field/Galois Theory Summary 4 2.1 Algebraic Field Extensions .................................................................................. 4 2.2 Automorphisms and the Galois Group ................................................................ 8 2.3 Cyclotomic Fields ................................................................................................ 10 2.4 Finite Fields ......................................................................................................... 11 2.5 The Galois Correspondence Theorem ................................................................... 12 2.6 The Discriminant ................................................................................................... 14 3 Rings of Algebraic Integers 16 3.1 Algebraic Integers ................................................................................................ 16 3.2 The Trace and Norm ............................................................................................. 20 3.3 The Discriminant ................................................................................................... 22 3.4 The Kronecker-Weber Theorem for Quadratic Extensions ................................ 29 3.5 Dedekind Domains ................................................................................................ 31 4 Splitting of Primes 36 4.1 Introduction ............................................................................................................ 36 4.2 Ramification Indices and Inertial Degrees .......................................................... 37 4.3 Splitting of Primes in Normal Extensions .......................................................... 43 4.4 Ramification and the Discriminant ...................................................................... 45 4.5 The Different ......................................................................................................... 46 5 Decomposition, Inertia, and Ramification Groups 49 5.1 Introduction ............................................................................................................ 49 v 5.2 The Main Result ................................................................................................... 51 5.3 Some Consequences of the Main Result ............................................................. 55 5.4 Splitting of Primes in Cyclotomic Fields ............................................................. 58 5.5 Ramification Groups ............................................................................................. 60 5.6 Hilbert's Formula ................................................................................................... 67 6 The Kronecker-Weber Theorem 70 6.1 Introduction ............................................................................................................ 70 6.2 Special Case: The field K has prime power degree pm over Q and p e Z is the only ramified prime ............................................................................. 71 6.3 Special Case: The field K has prime power degree over Q ................................. 78 6.4 General Case ......................................................................................................... 81 6.5 Examples .............................................................................................................. 84 Bibliography 87 vi List of Figures 2.1 The Galois correspondence for Q(^/2, i)/Q ....................................................... 14 5.1 The Galois correspondence for Q(v% V^VQ ................................................ 50 5.2 The tower of fields in the proof of Prop. 5.3.1 .................................................. 56 5.3 Left: The tower of fields in the proof of Prop. 5.3.2; Right: The corresponding tower of primes lying over P ............................................................................. 57 6.1 The tower of fields in the proof of Prop. 6.2.1 .................................................. 72 6.2 The tower of fields in the proof of Prop. 6.2.5 .................................................. 77 6.3 Left: The tower of fields in the proof of Prop. 6.3.1 along with the corre­ sponding degrees; Right: The corresponding tower of primes lying over q and their ramification indices. ............................................................................... 79 vii Acknowledgements I would like to thank my advisor, Prof. Elizabeth Allman, for working with me these past few years, even when she was out of the country. I would like to the thank the rest of my committee, Profs. John Rhodes and Jill Faudree, for reading and editing my thesis. I would also like to thank my wife, Kristen, for keeping me motivated and listening to me complain. Lastly, I would like to thank my pet turtle, Luna, for keeping me company during so many sleepless nights. 1 Chapter 1 Introduction A field extension K of Q is finite if K is finite dimensional as a Q-vector space; K is abelian if K/Q is Galois and the Galois group Gal(K/Q) is abelian. We will define and discuss the Galois group in Chapter 2. A cyclotomic field is a field obtained by adjoining roots of unity to Q. For example, Q(v/2) is a finite, abelian extension of Q. Let w8 = e2ni/8 = ^ + i-^22. Then w8 is a primitive 8-th root of unity, so Q(w8) is the 8-th cyclotomic field. Moreover, , 7 _ (V2 .V2\ (V2 ,V2\ _ R = ( t - + + - t~^)=y/2- Thus, \/2 e Q(w8), and so we must have Q(\/2) C Q(w8). This is an example of a more general classification result from Algebraic Number Theory called the Kronecker-Weber Theorem. Theorem (Kronecker-Weber). Every finite, abelian extension Qof is contained in a cyclo­ tomic field. According to [1], the Kronecker-Weber Theorem was originally proven by Leopold Kro- necker in 1853. However, his proof had some gaps in the case when the degree of the extension was a power of 2. The first accepted proof of this result was due to Heinrich Martin Weber in 1886. However, his proof also had a gap when the degree of the extension was 2, although this error went unnoticed for 90 years. The first correct proof was due to David Hilbert in 1896. Hilbert’s strategy of the proof eventually led to the field of mathe­ matics called Class Field Theory, which is the study of finite, abelian extensions of arbitrary fields. A problem that is still open in Class Field Theory is Hilbert’s Twelfth Problem, a generalization of the Kronecker-Weber Theorem. Open Problem (Hilbert’s Twelfth Problem). Extend the Kronecker-Weber Theorem to finite, abelian extensions of arbitrary fields. Not only is the Kronecker-Weber Theorem surprising, its proof is truly amazing. The idea of the proof is that for a finite, Galois extension
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