Theory of Algebraic Numbers (eBook)

Harold G. Diamond, Harry Pollard (Autoren)

eBook Download: EPUB

2012
192 Seiten
Dover Publications (Verlag)
978-0-486-15437-4 (ISBN)

Infos

Excellent intro to basics of algebraic number theory. Gausian primes; polynomials over a field; algebraic number fields; algebraic integers and integral bases; uses of arithmetic in algebraic number fields; more. 1975 edition.

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Detailed proofs and clear-cut explanations provide an excellent introduction to the elementary components of classical algebraic number theory in this concise, well-written volume.The authors, a pair of noted mathematicians, start with a discussion of divisibility and proceed to examine Gaussian primes (their determination and role in Fermat's theorem); polynomials over a field (including the Eisenstein irreducibility criterion); algebraic number fields; bases (finite extensions, conjugates and discriminants, and the cyclotomic field); and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture (concluding with discussions of Pythagorean triples, units in cyclotomic fields, and Kummer's theorem). In addition to a helpful list of symbols and an index, a set of carefully chosen problems appears at the end of each chapter to reinforce mathematics covered. Students and teachers of undergraduate mathematics courses will find this volume a first-rate introduction to algebraic number theory.

Inhalt

Chapter I. Divisibility 1. Uniqueness of factorization 2. A general problem 3. The Gaussian integers ProblemsChapter II. The Gaussian Primes 1. Rational and Gaussian primes 2. Congruences 3. Determination of the Gaussian primes 4. Fermat's theorem for Gaussian primes ProblemsChapter III. Polynomials over a field 1. The ring of polynomials 2. The Eisenstein irreducibility criterion 3. Symmetric polynomials ProblemsChapter IV. Algebraic Number Fields 1. Numbers algebraic over a field 2. Extensions of a field 3. Algebraic and transcendental numbers ProblemsChapter V. Bases 1. Bases and finite extensions 2. Properties of finite extensions 3. Conjugates and discriminants 4. The cyclotomic field ProblemsChapter VI. Algebraic Integers and Integral Bases 1. Algebraic integers 2. The integers in a quadratic field 3. Integral bases 4. Examples of integral bases ProblemsChapter VII. Arithmetic in Algebraic Number Fields 1. Units and primes 2. Units in a quadratic field 3. The uniqueness of factorization 4. Ideals in an algebraic number field ProblemsChapter VIII. The Fundamental Theorem of Ideal Theory 1. Basic properties of ideals 2. The classical proof of the unique factorization theorem 3. The modern proof ProblemsChapter IX. Consequences of the Fundamental Theorem 1. The highest common factor of two ideals 2. Unique factorization of integers 3. The problem of ramification 4. Congruences and norms 5. Further properties of norms ProblemsChapter X. Ideal Classes and Class Numbers 1. Ideal classes 2. Class numbers ProblemsChapter XI. The Fermat Conjecture 1. Pythagorean triples 2. The Fermat conjecture 3. Units in cyclotomic fields 4. Kummer's theorem Problems References; List of symbols; Index

Produkt-Details

• Erscheint lt. Verlag: 12.7.2012
• Reihe/Serie: Dover Books on Mathematics
• Sprache: englisch
• Maße: 140 x 140 mm
• Gewicht: 213 g
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Schlagworte: advanced math • Algebra • algebraic integers • Bases • class numbers • Congruences • Conjugates • Cyclotomic Fields • discriminants • divisibility • Education • eisenstein irreducibility criterion • fermat conjecture • fermats theorem • finite extensions • gaussian primes • ideal classes • ideal theory • integral bases • kummers theorem • Math • Mathematics • Nonfiction • Number Fields • Number Theory • polynomials • proofs • Pythagorean triples • Reference • Symmetric functions • Textbook • uft • undergraduate math
• ISBN-10: 0-486-15437-8 / 0486154378
• ISBN-13: 978-0-486-15437-4 / 9780486154374