Graduate text designed to prepare student for further study in the theory of fields, especially in algebraic number theory and class field theory. Galois theory and theory of valuations examined; special attention to development of infinite Galois theory, also detailed analysis of prolongation of rank-one valuations. "...clear, unsophisticated and direct..." — Math. Reviews. Over 200 exercises. Bibliography.
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3 of 3 people found the following review helpful:
5.0 out of 5 stars
Galois theory and some commutative algebra,
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This review is from: Algebraic Extensions of Fields (Paperback)
The book starts with a very clear presentation of the principles of Galois theory in two chapters: "Algebraic extensions" and "Galois theory", compareble to Artins short book Galois Theory: Lectures Delivered at the University of Notre Dame (Notre Dame Mathematical Lectures, Number 2).
The next three chapters are in essence about algebraic number fields, although he only defines these objects in an exercise in the last chapter. The approach is is a nice alternative to standard texts and goes via valuation theory. McCarthy first deals with valuation theory of fields. Then he deals with extensions of valuated fields. These two chapters are straight forward. In the last chapter he defines structures he calls "Dedekind fields", this concept is based on valuations and is a generalization of the concept of an algebraic number field. He then proceeds by proving the unique factorization theorem on the ideals of the rings in these fields and gives an introduction to Galois extensions of these Dedekind fields (read: algebraic number fields). The book contains more than 200 exercises many of which are challenging. All in all a very nice book.
2 of 2 people found the following review helpful:
5.0 out of 5 stars
Very concise, great for those who are ready,
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This review is from: Algebraic Extensions of Fields (Paperback)
The reviewer who gave this two stars sounds like he did not have sufficient preparation with the basics of abstract algebra. Yes, it is true, McCarthy brings in concepts like isomorphisms, fields, modding out a ring by an ideal, etc. without a definition and using properties of them without stating what he's using. But if you've taken an introduction to algebra class, and are now perhaps taking a following course in Galois theory - or if you're ready for higher level stuff like valuations on fields - this is a superb book for you. His conciseness is much like Rudin's in that you'll often find yourself asking, "How did this step work?", and you'll have to get out some paper and work it out, but that only enhances one's understanding of it. This book is dirt cheap, goes very far into advanced topics, and is excellent in its exposition.
1 of 4 people found the following review helpful:
2.0 out of 5 stars
Almost an example of how not to write a Math text?,
By R. Bagula "Roger L. Bagula" (Lakeside, Ca United States) - See all my reviews (VINE VOICE) (REAL NAME)
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This review is from: Algebraic Extensions of Fields (Paperback)
In the Preface he says he will be using Zorn's Lemma in proof and never any where states the Lemma.
His first statement is " Let k be a field"; he never gives a formal definition of a field. I shouldn't maybe gripe because I got this copy ( a few pennies and postage), because it is a plague on the market as books go? As another reviewer points out, it has some good and unique aspects, like a discussion of Dedekind fields and of Eisenstein polynomials. The major problem is that you need three other books to read this one :to explain the terms, to explain the notation like the divisors sums and products, to explain his definitions... Mostly he assumes too much of the reader or student. This book makes me very sorry for those who study Mathematics in the British or European systems: one shouldn't need a stack of texts to translate a text. I have read worse, but not many. I really hate writing reviews like this, but since the other review is 5 stars, there really has to be a reality check here.
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