By Waclaw Sierpinski

Best number theory books

Number Theory 1: Fermat's Dream

This can be the English translation of the unique eastern publication. during this quantity, "Fermat's Dream", middle theories in sleek quantity idea are brought. advancements are given in elliptic curves, $p$-adic numbers, the $\zeta$-function, and the quantity fields. This paintings provides a sublime standpoint at the ask yourself of numbers.

The Riemann Zeta-Function (De Gruyter Expositions in Mathematics)

The purpose of the sequence is to provide new and demanding advancements in natural and utilized arithmetic. good confirmed locally over twenty years, it deals a wide library of arithmetic together with numerous vital classics. The volumes offer thorough and designated expositions of the tools and concepts necessary to the subjects in query.

Algebraic Number Theory: Proceedings of an Instructional Conference Organized by the London Mathematical Society (A Nato Advanced Study Institute W)

This e-book offers a brisk, thorough therapy of the rules of algebraic quantity thought on which it builds to introduce extra complicated themes. all through, the authors emphasize the systematic improvement of thoughts for the categorical calculation of the fundamental invariants akin to earrings of integers, category teams, and devices, combining at each one degree concept with specific computations.

Additional resources for 250 problems in elementary number theory

Sample text

Rn are fields. R1 ⊕ R2 ⊕ · · · ⊕ Rn , 5. D. E. Rowe, Gauss, Dirichlet and the law of biquadratic reciprocity, The Mathematical Intelligencer 10 (1988), 13–26. This paper gives a discussion of the relationship between Gauss and Dirichlet, mainly concerning their contributions to number theory including their work on biquadratic reciprocity. 6. W. C. Waterhouse, Quadratic polynomials and unique factorization, American Mathematical Monthly 109 (2002), 70–72. 2 is taken from this paper. Biographies 1.

6 Sums and Products of Ideals 21 a = 0 so bd = 1. Thus b is a unit and J = b = D ⊃ J , a contradiction. Hence I is maximal. 6 Sums and Products of Ideals In this section we show how to add and multiply ideals to obtain further ideals. First we define the sum of two ideals. 1 (Sum of ideals) Let I and J be ideals in an integral domain D. The sum of I and J , written I + J , is defined by I + J = {i + j | i ∈ I, j ∈ J }. It is readily checked that I + J is also an ideal and that it is the minimal ideal containing both I and J .

CB609-01 CB609/Alaca & Williams August 7, 2003 17:16 Char Count= 0 Exercises 23 Proof: We show first that P ∩ D is an ideal of D. Let a, b ∈ P ∩ D. Then a, b ∈ P and a, b ∈ D. From the first of these, as P is an ideal, we see that a + b ∈ P. From the second, as D is an integral domain, it is closed under addition so that a + b ∈ D. Hence a + b ∈ P ∩ D. Now suppose that a ∈ P ∩ D and d ∈ D. As d ∈ D, a ∈ P and P is an ideal of D, we deduce that da ∈ P. As d ∈ D, a ∈ D and D being an integral domain is closed under multiplication, we see that da ∈ D.