A Course In Algebraic Number Theory

An introduction to algebraic number theory, covering both global and local fields. The prerequisite is a standard graduate course in algebra.

**Tag(s):**
Algebra

**Publication date**: 17 Jun 2010

**ISBN-10**:
0486477541

**ISBN-13**:
9780486477541

**Paperback**:
126 pages

**Views**: 30,253

A Course In Algebraic Number Theory

An introduction to algebraic number theory, covering both global and local fields. The prerequisite is a standard graduate course in algebra.

Terms and Conditions:

Book Excerpts:

This is a text for a basic course in algebraic number theory, written in accordance with the following objectives:

1. Provide reasonable coverage for a one-semester course.

2. Assume as prerequisite a standard graduate course in algebra, but cover integral extensions and localization before beginning algebraic number theory. For general algebraic background, see author's online text "Abstract Algebra: The Basic Graduate Year". The abstract algebra material is referred to in this text as TBGY.

3. Cover the general theory of factorization of ideals in Dedekind domains, as well as the number field case.

4. Do some detailed calculations illustrating the use of Kummer's theorem on lifting of prime ideals in extension fields.

5. Give enough details so that the reader can navigate through the intricate proofs of the Dirichlet unit theorem and the Minkowski bounds on element and ideal norms.

6. Cover the factorization of prime ideals in Galois extensions.

7. Cover local as well as global fields, including the Artin-Whaples approximation theorem and Hensel's lemma.

Robert B. Ash wrote:(C) Copyright 2003, by Robert B. Ash. Paper or electronic copies for noncommercial use may be made freely without explicit permission of the author. All other rights are reserved.

Book Excerpts:

This is a text for a basic course in algebraic number theory, written in accordance with the following objectives:

1. Provide reasonable coverage for a one-semester course.

2. Assume as prerequisite a standard graduate course in algebra, but cover integral extensions and localization before beginning algebraic number theory. For general algebraic background, see author's online text "Abstract Algebra: The Basic Graduate Year". The abstract algebra material is referred to in this text as TBGY.

3. Cover the general theory of factorization of ideals in Dedekind domains, as well as the number field case.

4. Do some detailed calculations illustrating the use of Kummer's theorem on lifting of prime ideals in extension fields.

5. Give enough details so that the reader can navigate through the intricate proofs of the Dirichlet unit theorem and the Minkowski bounds on element and ideal norms.

6. Cover the factorization of prime ideals in Galois extensions.

7. Cover local as well as global fields, including the Artin-Whaples approximation theorem and Hensel's lemma.

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About The Author(s)

Robert B. Ash (1935-2015) is Professor Emeritus of Mathematics in the Department of Mathematics at the University of Illinois. He was an Assistant Professor in Electrical Engineering at Columbia University and then a Visiting Assistant Professor at the University of California, Berkeley, before coming to the University of Illinois Department of Mathematics in 1963. He retired in 1990. Bob wrote thirteen textbooks but he was most proud of the eight books he posted on his web page during his retirement that were available for free downloading.

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