# A Classical Introduction to Modern Number Theory by Kenneth Ireland, Michael Rosen

By Kenneth Ireland, Michael Rosen

This well-developed, obtainable textual content information the historic improvement of the topic all through. It additionally offers wide-ranging assurance of vital effects with relatively uncomplicated proofs, a few of them new. This moment variation includes new chapters that offer an entire evidence of the Mordel-Weil theorem for elliptic curves over the rational numbers and an summary of modern growth at the mathematics of elliptic curves.

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N. Thus Ui is a unit for each i. Conversely, if Ui is a unit for each i, then U = (Ul' U2' ... , un) is a unit. For a ring R we denote the group ofunits by U(R). •. x U(Rn) is the set of n-tuples (Ul' U2' ... , un), where Ui ERi. This is a group under component-wise multiplication. 1. IfS = R 1 EB R 2 EB ... • x U(R n). X U(R 2) Let mI' m2' ... , mt be pairwise relatively prime integers. t/li will denote the natural homomorphism from Z to Z/miZ. We construct a map t/I from Z to Z/m 1 ZEBZ/m 2ZEB···EBZ/mt Z as folIows: t/I(n) = (t/ll(n), t/lin), ...

Am} is called a complete set ofresidues modulo m. For example, {O, 1,2, 3}, {4,9, 14, -1}, and {O, 1, - 2, -1} are complete sets of residues modulo 4. The set Z/mZ can be made into a ring by defining in a natural way addition and multiplication. This is accomplished by means of the following proposition. 3. lf a == c (m) and b == d (m), then a ab == cd (m). PROOF. (a If mic - a and mld - b, then ml(c - a) + b). Thus a + b == c + d (m). Notice that cd - ab = c(d - b) ~~ + b(c - + b == c + d (m) + (d - b) = (c and + d) - a).

Let N be the number of solutions to f(x) == O(n) and Ni be the number of solutions to f(x) == O(Pf'). Prove that N = N IN 2··· N,. 19. If p is an odd prime, show that 1 and -1 are the only solutions to x 2 == 1 (p"). 20. Show that x 2 == 1 (2 b) has one solution if b = 1, two solutions if b = 2, and four solutions if b ~ 3. 21. Use Exercises 18-20 to tind the number of solutions to x 2 == 1 (n). 22. Formulate and prove the Chinese Remainder Theorem in a principal ideal domain. 23. Extend the notion of congruence to the ring Z[i] and prove that a congruent to O or 1 modulo 1 + i.