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## Über dieses Buch

Introduction to Cyclotomic Fields is a carefully written exposition of a central area of number theory that can be used as a second course in algebraic number theory. Starting at an elementary level, the volume covers p-adic L-functions, class numbers, cyclotomic units, Fermat's Last Theorem, and Iwasawa's theory of Z_p-extensions, leading the reader to an understanding of modern research literature. Many exercises are included.
The second edition includes a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture. There is also a chapter giving other recent developments, including primality testing via Jacobi sums and Sinnott's proof of the vanishing of Iwasawa's f-invariant.

## Inhaltsverzeichnis

### Chapter 1. Fermat’s Last Theorem

Abstract
We start with a special case of Fermat’s Last Theorem, since not only was it the motivation for much work on cyclotomic fields but also it provides a sampling of the various topics we shall discuss later.
Lawrence C. Washington

### Chapter 2. Basic Results

Abstract
In this chapter we prove some basic results on cyclotomic fields which will lay the groundwork for later chapters. We let ζ n denote a primitive nth root of unity. First we determine the riqng of integers and discriminant of $$\mathbb{Q}$$n). We start with the prime power case.
Lawrence C. Washington

### Chapter 3. Dirichlet Characters

Abstract
In this chapter we introduce the basic facts about Dirichlet characters. We then show how they may be used to obtain information about the arithmetic of number fields. As a result, we show how to obtain ideal class groups containing prescribed subgroups.
Lawrence C. Washington

### Chapter 4. Dirichlet L-series and Class Number Formulas

Abstract
In this chapter we review some of the basic facts about L-series. Then their values at negative integers are given in terms of generalized Bernoulli numbers. Finally, we discuss the values at 1 and relations with class numbers.
Lawrence C. Washington

### Chapter 5. p-adic L-functions and Bernoulli Numbers

Abstract
In this chapter we shall construct p-adic analogues of Dirichlet L-functions. Since the usual series for these functions do not converge p-adically, we must resort to another procedure. The values of $$L\left( {s,\chi } \right)$$ at negative integers are algebraic, hence may be regarded as lying in an extension of $${\mathbb{Q}_p}$$. We therefore look for a p-adic function which agrees with $$L\left( {s,\chi } \right)$$ at the negative integers. With a few minor modifications, this is possible.
Lawrence C. Washington

### Chapter 6. Stickelberger’s Theorem

Abstract
The aim of this chapter is to give, for any abelian number field, elements of the group ring of the Galois group which annihilate the ideal class group.hey will form the Stickelberger ideal. The proof involves factoring Gauss sums as products of prime ideals, and since Gauss sums generate principal ideals, we obtain relations in the ideal class group. As an application, we prove Herbrand’s theorem which relates the nontriviality of certain parts of the ideal class group of $$\mathbb{Q}({\zeta _p})$$ to p dividing corresponding Bernoulli numbers. Then we calculate the index of the Stickelberger ideal in the group ring for $$\mathbb{Q}({\zeta _{{p^n}}})$$ and find it equals the relative class number. Finally, we prove a result, essentially due to Eichler, on the first case of Fermat’s Last Theorem. In the next chapter we shall use Stickelberger elements to give Iwasawa’s construction of p-adic L-functions.
Lawrence C. Washington

### Chapter 7. Iwasawa’s Construction of p-adic L-functions

Abstract
Following Iwasawa, we show how Stickelberger elements may be used to construct p-adic L-functions. The result yields a very useful representation of these functions in terms of a power series. As an application, we obtain information about the behavior of the p-part of the class number in a cyclotomic $${\mathbb{Z}_p}$$ -extension and prove that the Iwasawa μ-invariant vanishes for abelian number fields. Also, we show how many of the formulas we obtain have analogues in the theory of function fields over finite fields.
Lawrence C. Washington

### Chapter 8. Cyclotomic Units

Abstract
The determination of the unit group of an algebraic number field is rather difficult in general. However, for cyclotomic fields, it is possible to give explicitly a group of units, namely the cyclotomic units, which is of finite index in the full unit group. Moreover, this index is closely related to the class number, a fact which allows us to prove Leopoldt’s p-adic class number formula. Finally, we study more closely the units of the pth cyclotomic field, and give relations with p-adic L-functions and with Vandiver’s conjecture.
Lawrence C. Washington

### Chapter 9. The Second Case of Fermat’s Last Theorem

Abstract
In Chapters 1 and 6 we treated the first case of Fermat’s Last Theorem, showing that there are no solutions provided certain conditions are satisfied by the class number.
Lawrence C. Washington

### Chapter 10. Galois Groups Acting on Ideal Class Groups

Abstract
Relatively recently, it has been observed, in particular by Iwasawa and Leopoldt, that the action of Galois groups on ideal class groups can be used to great advantage to reinterpret old results and to obtain new information on the structure of class groups. In this chapter we first give some results which are useful when working with class groups and class numbers. We then present the basic machinery, essentially Leopoldt’s Spiegelungssatz, which underlies the rest of the chapter. As applications, Kummer’s result “ $$p\left| {{h^ + } \Rightarrow p} \right|{h^ - }$$ ” is made more precise and a classical result of Scholz on class groups of quadratic fields is proved. Finally, we show that Vandiver’s conjecture implies that the ideal class group of $$\mathbb{Q}\left( {{\zeta _{{p^n}}}} \right)$$is isomorphic to the minus part of the group ring modulo the Stickelberger ideal.
Lawrence C. Washington

### Chapter 11. Cyclotomic Fields of Class Number One

Abstract
In this chapter we determine those m for which $$\mathbb{Q}({\zeta _m})$$ has class number one. In Chapter 4, the Brauer–Siegel theorem was used to show that there are only finitely many such fields, but the result was noneffective: there was no computable bound on m. So we need other techniques. Since hn divides h m if n divides m, it is reasonable to start with m prime. In 1964 Siegel showed that hp = 1 implies pC, where C is a computable constant, but the constant was presumably too large to make computations feasible. In 1971, Montgomery and Uchida independently obtained much better values of C, from which it followed that $${h_p} = 1 \Leftrightarrow p \leqslant 19$$ Masley was then able to use this information, plus a table of h m - for ø(m) ≤ 256, to explicitly determine all m with hm=1.
Lawrence C. Washington

### Chapter 12. Measures and Distributions

Abstract
The concept of a distribution, as given in this chapter, is one that occurs repeatedly in mathematics, especially in the theory of cyclotomic fields. As we shall see, many ideas from Chapters 4, 5, 7, and 8 fit into this general framework. The related concept of a measure yields a p-adic integration theory which allows us to interpret the p-adic L-function as a Mellin transform, as in the classical case.
Lawrence C. Washington

### Chapter13. Iwasawa’s Theory of ℤ-extensions

Abstract
The theory of ℤ-extensions has turned out to be one of the most fruitful areas of research in number theory in recent years. The subject receives its motivation from the theory of curves over finite fields, which is known to have a strong analogy with the theory of number fields. In the case of curves, it is convenient to extend the field of constants to its algebraic closure, which amounts to adding on roots of unity. There is a natural generator of the Galois group, namely the Frobenius, and its action on various modules yields zeta functions and L-functions. In the number field case, it turns out to be too unwieldy, at least at present, to use all roots of unity. Instead, it is possible to obtain a satisfactory theory by just adjoining the p-power roots of unity for a fixed prime p This yields a ℤ-extension. The action of a generator of the Galois group on a certain module yields, at least conjecturally, the p-adic L-functions.
Lawrence C. Washington

### Chapter 14. The Kronecker—Weber Theorem

Abstract
The Kronecker—Weber theorem asserts that every abelian extension of the rationals is contained in a cyclotomic field. It was first stated by Kronecker in 1853, but his proof was incomplete. In particular, there were difficulties with extensions of degree a power of 2. Even in the proof we give below this case requires special consideration. The first proof was given by Weber in 1886 (there was still a gap; see Neumann [1]). Both Kronecker and Weber used the theory of Lagrange resolvents. In 1896, Hilbert gave another proof which relied more on an analysis of ramification groups. Now, the theorem is usually given as an easy consequence of class field theory. We do this in the Appendix. The main point is that in an abelian extension the splitting of primes is determined by congruence conditions, and we already know that p splits in $$\mathbb{Q}\left( {{\zeta _n}} \right)$$ if $$p \equiv 1$$ and only if mod n.
Lawrence C. Washington

### Chapter 15. The Main Conjecture and Annihilation of Class Groups

Abstract
In the mid 1980s, Thaine and Kolyvagin invented new techniques for constructing relations in ideal class groups. These methods have had profound consequences. Not only is it now possible to give a fairly elementary proof of the Main Conjecture, but these ideas also allowed Rubin to give the first examples of finite Tate-Shafarevich groups for elliptic curves.
Lawrence C. Washington

### Chapter 16. Miscellany

Abstract
Suppose n is a large odd number that we want to test for primality. A standard procedure is to compute, for example, 2n-1 (mod n). If the answer is not 1 (mod n), then n is composite, and if the answer is 1 (mod n), we suspect n might be prime.
Lawrence C. Washington

### Backmatter

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