The Theory of Algebraic NumbersCourier Corporation, 12 juil. 2012 - 192 pages 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. |
Table des matières
1 | |
THE GAUSSIAN PRIMW | 14 |
POLYNOMIALS OVER A FIELD | 25 |
ALGEBRAIC NUMBER FIELDS | 44 |
BASES | 59 |
ALGEBRAIC INTEGERS AND INTEGRAL BASES | 74 |
ARITHMETIC IN ALGEBRAIC NUMBER FIELDS | 88 |
THE FUNDAMENTAL THEOREM OF IDEAL THEORY | 102 |
CONSEQUENCES OF THE FUNDAMENTAL THEOREM | 120 |
IDEAL CLASSES AND CLASS NUMBERS | 139 |
THE FERMAT CONJECTURE | 146 |
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Expressions et termes fréquents
a₁ algebraic integer algebraic number field algebraic over F B₁ Bibliography CALCULUS Chapter class number Classic complete residue system congruence conjugates contains Corollary degree denote DIFFERENTIAL EQUATIONS divides divisible divisor element example exist extension of F factorization of integers Fermat's field F Find finite extension finite number follows form a basis fundamental theorem Gaussian integers Gaussian primes Hence integral basis INTRODUCTION irreducible Let f(x linearly independent MATHEMATICAL maximal ideal minimal polynomial modulo monic nomial non-zero ideal Number Theory P₁ polynomial over F positive integer prime ideals prime in G prime number primitive problems proof of Theorem quadratic field rational integers rational numbers rational prime relatively prime residue class root of unity satisfies Show solution suppose symmetric polynomial Theorem 4.6 totality of integers transcendental unique factorization unit w₁ zero απ