The eld l kq m is a compositum of galois extensions of q, and is therefore galois over q with gallq isomorphic to a subgroup of galkq galq. The kronecker weber theorem and concepts in algebraic number. The kroneckerweber tells us what abelian extensions of the rational numbers are like. Kronecker stated this theorem in 1853, but his proof was incomplete. The key tool in the proof is that there is only one cyclic extension of q of degree p, p an odd prime, where p is the only ramified prime. Prices in represent the retail prices valid in germany unless otherwise indicated. The kronecker weber theorem asserts that every abelian extension of the rationals is contained in a cyclotomic field. While a complete proof of the fundamental theorem of galois. This was the rst correct complete proof of the theorem.
Prices do not include postage and handling if applicable. In this paper we present a combinatorial proof of the kronecker weber theorem for global fields of positive characteristic. E0,1728so the theorem holds true for those two cases aswell. After that, the reader willing to believe occasional facts could probably read the remaining chapters randomly. Primes in arithmetic progressions, infinite products, partial summation and dirichlet series, dirichlet characters, l1, x and class numbers, the distribution of the primes, the prime number theorem, the functional equation, the prime number theorem for arithmetic progressions, siegels. Kroneckerweberhilbert every abelian extension of the rational numbers q is contained in a cyclotomic extension. It is a vast generalization of the fact we proved in a previous lecture.
Part of the graduate texts in mathematics book series gtm, volume 83. Shafarevich, a new proof of the kroneckerweber theorem, collection of articles. Recall the kroneckerweber theorem, which identifies the maximal abelian extension of q. In the global case there is a long history of elementary proofs. Representation theory of finite abelian groups over c 17 5. What makes marcus book particularly unusual and compelling is the. The kroneckerweber theorem november 30, 2007 let us begin with the local statement. Elementary and analytic theory of algebraic numbers. David hilbert gave a new proof of kroneckers original. The kroneckerweber theorem says the converse also holds. If time permits, i will give another proof of the kroneckerweber theo rem. A combinatorial proof of the kroneckerweber theorem in positive characteristic.
We also present the necessary classical galois theory and algebraic number theory to understand the given proof, thus allowing students with a. Instead we rely on facts about the principal units in a finite abelian extension of k as a module for the galois group. Greenberg provided an elementary proof of the kroneckerweber theorem here amer. Pdf an elementary proof of the local kroneckerweber theorem. In algebraic number theory, the kroneckerweber theorem states that every finite abelian extension of the field of rational numbers q, or in other words, every algebraic number field whose galois group over q is abelian, is a subfield of a cyclotomic field, i.
Schappacher, on the history of hilberts twelfth problem, i. These are some brief notes on the famous kroneckerweber theorem, which says that cyclotomic extensions of q capture all abelian extension of q. K theorem implies the global kroneckerweber theorem. For nonabelian extensions, serious progress began only about 25 years ago. Prices in gbp apply to orders placed in great britain only. Let kq p be a cyclic extension of degree r for some prime 6 p. Introduction to the english edition of hilberts zahlbericht. The kronecker weber theorem and concepts in algebraic number theory a thesis submitted in partial ful llment of the requirements for the degree of master of science at george mason university by marla schnall bachelor of arts yale college, 1986 director. I was reading local kroneckerweber theorem implies global one in a course manual, but theres some parts i dont understand. The theorem of kronecker weber states that every abelian extension of. Recall that an abelian extension is a finite field extension kq such that the galois group galkq. Introduction these are some notes on the kroneckerweber theorem in algebraic number theory for math 2530 at brown university. The theorem was first stated by kronecker though his.
The kronecker weber theorem and concepts in algebraic. Olaf neumann, two proofs of the kroneckerweber theorem according to kronecker, and weber, j. A lower bound for the height in abelian extensions sciencedirect. The origins of class field theory lie in the quadratic reciprocity law proved by gauss. Given an abelian polynomial especially quadratic or cubic, how can we solve it in terms of one of these sums.
For each rami ed prime pof q, pick a prime pjpand let k p be the completion of kat p. The first part discusses elementary topics such as primes, factorization, continued fractions, and quadratic forms, in the context of cryptography, computation, and deep open research problems. Even in the proof we give below this case requires special consideration. The variety of these topics, and the occasional links and interplays. The kroneckerweber theorem is one of the earliest known results in class field theory. The kroneckerweber theorem characterizes abelian extensions of q. An argument in the lemma 4 was found to be wrong as noticed in correction to an elementary proof of the kroneckerweber theorem amer. The propositions in the book come from the realms of plane geometry, algebra, number theory, analysis, and the theory of sets. Local kroneckerweber theorem implies the global one. We give a new proof of the theorem of kroneckerweber based on kummer theory and stickelbergers theorem. To the sixtieth birthday of academician ivan matveevich vinogradov, trudy mat. David hilbert gave a new proof of kroneckers original statement in 1896 4. Weber gave a proof in 1886, but apparently there were still a gap in it. Neumann, two proofs of the kroneckerweber theorem according to kronecker, and weber, j.
The local kroneckerweber theorem implies the global kroneckerweber theorem. The theorem asserts that the compositum uln is the maximal abelian extension of k. A combinatorial proof of the kroneckerweber theorem in. Because of this some people call the result the kroneckerweber hilbert theorem. In 1853, kronecker announced what is now called the kroneckerweber theorem. The second is a simple application of the main results of class field theory, which classifies abelian extension of an arbitrary number field. The kroneckerweber theorem asserts that every abelian extension of the rationals is contained in a cyclotomic field. This article is cited in 1 scientific paper total in 1 paper a new proof of the kroneckerweber theorem i. We include here shafare vichs proof of the kroneckerweber theorem, and also the main properties of adeles and ideles. This paper is an introduction to the padic numbers and their eld extensions with the goal of proving the kroneckerweber theorem by using the local kroneckerweber theorem. An introductory course in elementary number theory.
Then k is contained in a cyclotomic extension of q p. In fact, the local and global versions are equivalent. Chapter 5 contains the fun damental notions of the theory of padic fields, and chapter 6 brings their applications to the study of algebraic number fields. Jay shapiro, professor department of mathematics spring semester 2014 george mason. The proof rests on elementary congruences in cyclotomic fields and on kroneckerweber theorem. Pdf an elementary proof of the kroneckerweber theorem. Representation theory of finite groups anupam singh. Basically i am looking for a proof of the kroneckerweber theorem that is constructive enough that i can compute with it. The main tools are the use of witt vectors and their arithmetic developed by h. Download an introductory course in elementary number theory download free online book chm pdf. The classical results that can be found in most books about algebraic number theory will usually be stated without proof, whereas the results that are more specific. Mar 01, 2014 the proof of this theorem uses artintakagi. The result is now known as the kroneckerweber theorem.
If time permits, i will give another proof of the kroneckerweber theorem. It was first stated by kronecker in 1853, but his proof was incomplete. In this approach, the theorem is deduced from the corresponding statement for local. It is reasonable to call this result the local kroneckerweber theorem. Department of mathematics the university of chicago. For example, the reader might skip directly to chapter to learn about zpextensions. Both the kroneckerweber theorem and the local version are most easily proved using class field theory.
In the classical case, possibly the simplest proof of the kroneckerweber theorem uses ramification groups see. The kroneckerweber theorem school of mathematics, tifr. Kroneckers proof, by his own admittance, had di culties with extensions of 2power degree. If time permits, i will give another proof of the kronecker weber theo rem. Further, for the sake of variety, i have sampled different branches of mathematics. Elementary number theory primes, congruences, and secrets. These concepts include rings of integers as dedekind domains, nite elds and residue extensions, rami ed primes. This is a textbook about classical elementary number theory and elliptic curves.
This paper considers a proof based on foundational concepts in algebraic number theory that was presented by greenberg in the 1970s. Analytic number theory lecture notes by andreas strombergsson. The kroneckerweber theorem summer school on cyclotomic. Lubin and tate have shown how to explicitly construct an abelian extension of k which they prove to be the maximal abelian. To prove the local kroneckerweber theorem it thus su ces to consider cyclic extensions kq pof prime power degree r. The global kroneckerweber theorem holds if and only if the local kroneckerweber theorem holds. Every nite abelian extension of q lies in a cyclotomic eld q m for some m. Webers proof was also incomplete but the gap was not noticed up to more than ninety years later by olaf neuman 3. Essentially, a complete proof of the kroneckerweber theorem using basic algebraic number. Essentially, a complete proof of the kroneckerweber theorem using basic algebraic number theory, galois theory and higher ramfication groups. The last chapter, on the kroneckerweber theorem, can be read after chapter 2. Proof see any introductory algebraic number theory book or s2 p. Notes on class field theory mathematics and statistics mcgill.
Galois theory, the study of the structure and symmetry of a polynomial or associated field extension, is a standard tool for showing the insolvability of a quintic equation by radicals. An introductory course in elementary number theory download. Kroneckerweber hayes theorem bibliography number elds henrich weber provided a proof of kroneckers result in 1886 3. The first complete proof of the kroneckerweber theorem. The kroneckerweber theorem a bachelor thesis written by. Analog of the kroneckerweber theorem in positive characteristic villasalvador, gabriel d. The generalization took place as a longterm historical project, involving quadratic forms and their genus theory, work of ernst kummer and leopold kroneckerkurt hensel on ideals and completions, the theory of cyclotomic and kummer extensions the first two class field theories were very explicit. The kroneckerweber theorem states that all abelian extensions are sub elds of cy clotomic elds. Etorsionis the set ofxcoordinates of all the torsionpoints of e. A number field extension is said to be unramified if all prime ideals are unramified. A simple proof of the kroneckerweber theorem william stein. In particular, there were difficulties with extensions of degree a power of 2.
Analogue of the kronecker weber theorem in positive. In this paper we present a combinatorial proof of the kroneckerweber theorem for global fields of positive characteristic. Assuming the kroneckerweber theorem, we can deduce strong results about. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Kqp is a finite abelian extension of local fields, then, by corollary 11. The generalization took place as a longterm historical project, involving quadratic forms and their genus theory, work of ernst kummer and leopold kronecker kurt hensel on ideals and completions, the theory of cyclotomic and kummer extensions. In this context a field is a collection of elements with two binary operations on them, addition and multiplicatio. Kroneckerweber every abelian extension of q is contained in a cyclotomic ex.
185 456 1340 265 945 371 1061 394 551 1236 672 834 431 1167 111 546 1090 1037 1530 1429 779 1309 196 518 1239 1074 413 965 1317 1410 628 117 728