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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…mehr

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Produktbeschreibung
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.
• Produktdetails
• Graduate Texts in Mathematics Vol.83
• Verlag: Springer, Berlin
• 2. Aufl.
• Seitenzahl: 508
• Erscheinungstermin: 5. Dezember 1996
• Englisch
• Abmessung: 241mm x 161mm x 35mm
• Gewicht: 965g
• ISBN-13: 9780387947624
• ISBN-10: 0387947620
• Artikelnr.: 06672940
Inhaltsangabe
1 Fermat's Last Theorem.
2 Basic Results.
3 Dirichlet Characters.
4 Dirichlet L
series and Class Number Formulas.
5 p
functions and Bernoulli Numbers.
5.1. p
5.2. p
functions.
5.3. Congruences.
5.4. The value at s = 1.
5.5. The p
5.6. Applications of the class number formula.
6 Stickelberger's Theorem.
6.1. Gauss sums.
6.2. Stickelberger's theorem.
6.3. Herbrand's theorem.
6.4. The index of the Stickelberger ideal.
6.5. Fermat's Last Theorem.
7 Iwasawa's Construction of p
functions.
7.1. Group rings and power series.
7.2. p
functions.
7.3. Applications.
7.4. Function fields.
7.5. µ = 0.
8 Cyclotomic Units.
8.1. Cyclotomic units.
8.2. Proof of the p
adic class number formula.
8.3. Units of $$\mathbb{Q}\left( {{\zeta _p}} \right)$$ and Vandiver's conjecture.
8.4. p
9 The Second Case of Fermat's Last Theorem.
9.1. The basic argument.
9.2. The theorems.
10 Galois Groups Acting on Ideal Class Groups.
10.1. Some theorems on class groups.
10.2. Reflection theorems.
10.3. Consequences of Vandiver's conjecture.
11 Cyclotomic Fields of Class Number One.
11.1. The estimate for even characters.
11.2. The estimate for all characters.
11.3. The estimate for hm
.
11.4. Odlyzko's bounds on discriminants.
11.5. Calculation of hm+.
12 Measures and Distributions.
12.1. Distributions.
12.2. Measures.
12.3. Universal distributions.
13 Iwasawa's Theory of $${\mathbb{Z}_p}$$ extensions.
13.1. Basic facts.
13.2. The structure of A
modules.
13.3. Iwasawa's theorem.
13.4. Consequences.
13.5. The maximal abelian p
extension unramified outside p.
13.6. The main conjecture.
13.7. Logarithmic derivatives.
13.8. Local units modulo cyclotomic units.
14 The Kronecker
Weber Theorem.
15 The Main Conjecture and Annihilation of Class Groups.
15.1. Stickelberger's theorem.
15.2. Thaine's theorem.
15.3. The converse of Herbrand's theorem.
15.4. The Main Conjecture.
15.6. Technical results from Iwasawa theory.
15.7. Proof of the Main Conjecture.
16 Miscellany.
16.1. Primality testing using Jacobi sums.
16.2. Sinnott's proof that µ = 0.
16.3. The non
p
part of the class number in a $${\mathbb{Z}_p}$$ extension.
1. Inverse limits.
2. Infinite Galois theory and ramification theory.
3. Class field theory.
Tables.
1. Bernoulli numbers.
2. Irregular primes.
3. Relative class numbers.
4. Real class numbers.
List of Symbols.

1 Fermat's Last Theorem.- 2 Basic Results.- 3 Dirichlet Characters.- 4 Dirichlet L-series and Class Number Formulas.- 5 p-adic L-functions and Bernoulli Numbers.- 5.1. p-adic functions.- 5.2. p-adic L-functions.- 5.3. Congruences.- 5.4. The value at s = 1.- 5.5. The p-adic regulator.- 5.6. Applications of the class number formula.- 6 Stickelberger's Theorem.- 6.1. Gauss sums.- 6.2. Stickelberger's theorem.- 6.3. Herbrand's theorem.- 6.4. The index of the Stickelberger ideal.- 6.5. Fermat's Last Theorem.- 7 Iwasawa's Construction of p-adic L-functions.- 7.1. Group rings and power series.- 7.2. p-adic L-functions.- 7.3. Applications.- 7.4. Function fields.- 7.5. µ = 0.- 8 Cyclotomic Units.- 8.1. Cyclotomic units.- 8.2. Proof of the p-adic class number formula.- 8.3. Units of
$$\mathbb{Q}\left( {{\zeta _p}} \right)$$
and Vandiver's conjecture.- 8.4. p-adic expansions.- 9 The Second Case of Fermat's Last Theorem.- 9.1. The basic argument.- 9.2. The theorems.- 10 Galois Groups Actingon Ideal Class Groups.- 10.1. Some theorems on class groups.- 10.2. Reflection theorems.- 10.3. Consequences of Vandiver's conjecture.- 11 Cyclotomic Fields of Class Number One.- 11.1. The estimate for even characters.- 11.2. The estimate for all characters.- 11.3. The estimate for hm-.- 11.4. Odlyzko's bounds on discriminants.- 11.5. Calculation of hm+.- 12 Measures and Distributions.- 12.1. Distributions.- 12.2. Measures.- 12.3. Universal distributions.- 13 Iwasawa's Theory of
$${\mathbb{Z}_p} -$$
extensions.- 13.1. Basic facts.- 13.2. The structure of A-modules.- 13.3. Iwasawa's theorem.- 13.4. Consequences.- 13.5. The maximal abelian p-extension unramified outside p.- 13.6. The main conjecture.- 13.7. Logarithmic derivatives.- 13.8. Local units modulo cyclotomic units.- 14 The Kronecker-Weber Theorem.- 15 The Main Conjecture and Annihilation of Class Groups.- 15.1. Stickelberger's theorem.- 15.2. Thaine's theorem.- 15.3. The converse of Herbrand's theorem.- 15.4. The Main Conjecture.- 15.5. Adjoints.- 15.6. Technical results from Iwasawa theory.- 15.7. Proof of the Main Conjecture.- 16 Miscellany.- 16.1. Primality testing using Jacobi sums.- 16.2. Sinnott's proof that µ = 0.- 16.3. The non-p-part of the class number in a
$${\mathbb{Z}_p} -$$
extension.- 1. Inverse limits.- 2. Infinite Galois theory and ramification theory.- 3. Class field theory.- Tables.- 1. Bernoulli numbers.- 2. Irregular primes.- 3. Relative class numbers.- 4. Real class numbers.- List of Symbols.