Algebraic Number FieldsAmerican Mathematical Soc., 5 déc. 1995 The book is directed toward students with a minimal background who want to learn class field theory for number fields. The only prerequisite for reading it is some elementary Galois theory. The first three chapters lay out the necessary background in number fields, such as the arithmetic of fields, Dedekind domains, and valuations. The next two chapters discuss class field theory for number fields. The concluding chapter serves as an illustration of the concepts introduced in previous chapters. In particular, some interesting calculations with quadratic fields show the use of the norm residue symbol. For the second edition the author added some new material, expanded many proofs, and corrected errors found in the first edition. The main objective, however, remains the same as it was for the first edition: to give an exposition of the introductory material and the main theorems about class fields of algebraic number fields that would require as little background preparation as possible. Janusz's book can be an excellent textbook for a year-long course in algebraic number theory; the first three chapters would be suitable for a one-semester course. It is also very suitable for independent study. |
Table des matières
| 1 | |
COMPLETE FIELDS | 83 |
DECOMPOSITION GROUPS AND THE ARTIN MAP | 121 |
ANALYTIC METHODS AND RAY CLASSES | 135 |
CLASS FIELD THEORY | 169 |
QUADRATIC FIELDS | 233 |
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Expressions et termes fréquents
abelian extension abelian group absolute value algebraic integers algebraic number field Artin map basis Cauchy sequence Chapter class group class number coefficients computation conductor congruence subgroup contains COROLLARY coset cyclic group decomposition group Dedekind ring defined denote determine distinct prime divisible embedding equal equivalent Exercise extension of Q factorization finite number follows fractional ideal Frobenius automorphism Galois extension Galois group H₁ homomorphism ideal group implies infinite prime integral closure isomorphism kernel lattice Lemma matrix maximal ideal minimum polynomial modulus multiplicative NL/K nonarchimedean nonzero prime ideal norm positive integer prime ideal prime integer primitive principal ideal Proposition prove quadratic quotient field ramified real number reciprocity law holds relative degree relatively prime ring of algebraic root of unity Section set of primes splits completely subfield Suppose th root Theorem unit group unramified valuation ring
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Page 4 - ... is the determinant of the (n — 1) X (n — 1) matrix obtained by deleting the row and column in A in which a,, is located.