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Catalogue Information
Field name
Details
Dewey Class
512.7
Title
Cyclotomic Fields ([EBook]) / by Serge Lang.
Author
Lang, Serge , 19272005
Other name(s)
SpringerLink (Online service)
Publication
New York, NY : Springer , 1978.
Physical Details
253 pages : online resource.
Series
Graduate texts in mathematics
00725285 ; ; 59
ISBN
9781461299455
Summary Note
Kummer's work on cyclotomic fields paved the way for the development of algebraic number theory in general by Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. However, the success of this general theory has tended to obscure special facts proved by Kummer about cyclotomic fields which lie deeper than the general theory. For a long period in the 20th century this aspect of Kummer's work seems to have been largely forgotten, except for a few papers, among which are those by Pollaczek [Po], ArtinHasse [AH] and Vandiver [Va]. In the mid 1950's, the theory of cyclotomic fields was taken up again by Iwasawa and Leopoldt. Iwasawa viewed cyclotomic fields as being analogues for number fields of the constant field extensions of algebraic geometry, and wrote a great sequence of papers investigating towers of cyclotomic fields, and more generally, Galois extensions of number fields whose Galois group is isomorphic to the additive group of padic integers. Leopoldt concentrated on a fixed cyclotomic field, and established various padic analogues of the classical complex analytic class number formulas. In particular, this led him to introduce, with Kubota, padic analogues of the complex Lfunctions attached to cyclotomic extensions of the rationals. Finally, in the late 1960's, Iwasawa [Iw 1 I] . made the fundamental discovery that there was a close connection between his work on towers of cyclotomic fields and these padic Lfunctions of LeopoldtKubota.:
Contents note
1 Character Sums  1. Character Sums Over Finite Fields  2. Stickelbergerâ€™s Theorem  3. Relations in the Ideal Classes  4. Jacobi Sums as Hecke Characters  5. Gauss Sums Over Extension Fields  6. Application to the Fermat Curve  2 Stickelberger Ideals and Bernoulli Distributions  1. The Index of the First Stickelberger Ideal  2. Bernoulli Numbers  3. Integral Stickelberger Ideals  4. General Comments on Indices  5. The Index for k Even  6. The Index for k Odd  7. Twistings and Stickelberger Ideals  8. Stickelberger Elements as Distributions  9. Universal Distributions  10. The DavenportHasse Distribution  3 Complex Analytic Class Number Formulas  1. Gauss Sums on Z/mZ  2. Primitive Lseries  3. Decomposition of Lseries  4. The (Â±1)eigenspaces  5. Cyclotomic Units  6. The Dedekind Determinant  7. Bounds for Class Numbers  4 The padic Lfunction  1. Measures and Power Series  2. Operations on Measures and Power Series  3. The Mellin Transform and padic Lfunction  4. The padic Regulator  5. The Formal Leopoldt Transform  6. The padic Leopoldt Transform  5 Iwasawa Theory and Ideal Class Groups  1. The Iwasawa Algebra  2. Weierstrass Preparation Theorem  3. Modules over Zp[[X]]  4. Zpextensions and Ideal Class Groups  5. The Maximal pabelian pramified Extension  6. The Galois Group as Module over the Iwasawa Algebra  6 Kummer Theory over Cyclotomic Zpextensions  1. The Cyclotomic Zpextension  2. The Maximal pabelian pramified Extension of the Cyclotomic Zpextension  3. Cyclotomic Units as a Universal Distribution  4. The LeopoldtIwasawa Theorem and the Vandiver Conjecture  7 Iwasawa Theory of Local Units  1. The KummerTakagi Exponents  2. Projective Limit of the Unit Groups  3. A Basis for U(?) over A  4. The CoatesWiles Homomorphism  5. The Closure of the Cyclotomic Units  8 LubinTate Theory  1. LubinTate Groups  2. Formal padic Multiplication  3. Changing the Prime  4. The Reciprocity Law  5. The Kummer Pairing  6. The Logarithm  7. Application of the Logarithm to the Local Symbol  9 Explicit Reciprocity Laws  1. Statement of the Reciprocity Laws  2. The Logarithmic Derivative  3. A Local Pairing with the Logarithmic Derivative  4. The Main Lemma for Highly Divisible x and ? = xn  5. The Main Theorem for the Symbol ?x, xn?n  6. The Main Theorem for Divisible x and ? = unit  7. End of the Proof of the Main Theorems.
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Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site
http://dx.doi.org/10.1007/9781461299455
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Subject References:
Mathematics
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Number theory
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Authors:
Lang, Serge 19272005
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Lang, Serge. 19272005.
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Corporate Authors:
SpringerLink (Online service)
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Series:
Graduate texts in mathematics
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GTM
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Classification:
512.7
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