# A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84)

Language: English

Pages: 394

ISBN: 038797329X

Format: PDF / Kindle (mobi) / ePub

This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. This second edition contains two new chapters that provide a complete proof of the Mordel-Weil theorem for elliptic curves over the rational numbers and an overview of recent progress on the arithmetic of elliptic curves.

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generators are of the form g\ where (k, n) = I, n being the order of G. 14. Let A be a finite abelian group and a, b e A elements of order m and n, respectively . If (m, n) = I, prove that ab has order mn. 15. Let K be a field and G £; K* a finite subgroup of the multiplicative group of K. Extend the arguments used in the proof of Theorem I to show that G is cyclic. 16. Calculate the solutions to x J ;: I (19) and x 4 == I (17). 17. Use the fact that 2 is a primitive root modulo 29 to find the

{I, 2, . . . , (p - I)/2} and {ml' m 2, . . . , m(p -1)/2} coincide. Multiply the congruences l'a == ±m t (p), 2 ·a == ±m2 (p), ... , «p - I )/2)a == ± m(p_ 1)/2 (p). We obtain (p; I)! dP-I)/2 == (-I)I'(P ; I)! (p). This yields d P-1) /2 == (-1)~ (p). By Proposition 5.1.2, dP-I )/2 == (alp)(p). The result follows. 0 Gauss's lemma is an extremely powerful tool. We shall base our first proof of the quadratic reciprocity law on it. Before getting to that, however, 53 §2 Law of Quadratic

L xU)e = j n Ip -11/2 f. (e(2k - l l= - = (ePZ - e=IP-\2k-l))) I)h( e=). (3) k= I j = 1 The coefficient of :; (p - I )12 L"- I on the left-hand side of (3) is eas ily seen to be U) .(p - tr : 1)/ 2 1)/2 )«~ ~ 1~/2)! - )J f. (4k - P - 2). I On the other hand by Exercise 21 the coefficient of z ( p - 1)/2 on the right-hand side of(3) is pA IB where p,r B , A and B being integers. Equating coefficients, multiplying by B«p - 1)/2)! and reducing modulo p sho ws that (1) n p-

p - 2 with an ..error term" 2JP. This shows that for large primes p there are always many solutions. If p == I (3), there are always at least six solut ions since x 3 = 1 and i = 1 have three solutions each and we can write 1 + 0 = 1 and 0 + 1 = I. For p = 7 and 13 these are the only solut ions. For p = 19 other solutions exist ; e.g., 33 + 103 == 1 (19). These "nontrivial" solutions exist for all primes p ~ 19since it follows from the estimate that N p ~ p - 2 - 2JP > 6 forp~19. Using Jacobi

either as a formal power series or as a function of a complex variable defined and analytic on the disc {u E C/ Iu I < q- "}, It may seem strange to deal with Z iu) instead of directly considering the series Loo= t Nsu s. The reasons are mainly historical, although as we shall see the zeta funct ion is, in fact, easier to handle. See the remarks at the end of this section. As a first example, consider the hyperplane at infinity. By definit ion this is the set of points [ao,' .. ,an] E pn(F) with