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

By Kenneth Ireland, Michael Rosen

This well-developed, obtainable textual content info the historic improvement of the topic all through. It additionally presents wide-ranging insurance of vital effects with relatively simple proofs, a few of them new. This moment version includes new chapters that supply an entire evidence of the Mordel-Weil theorem for elliptic curves over the rational numbers and an summary of modern development at the mathematics of elliptic curves.

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If n > 1, Jl o /(n) = Ldln Jl(d) = O. The same proof works for / o Jl. O PROOF. Theorem 2 (M6bius Inversion Theorem). Let F(n) Ldln Jl(d)F(n/d). = Ldlnf(d). Thenf(n) F = f 0/. Thus F o Jl = (f 0/) o Jl = f o (l o Jl) = f that f(n) = F o Jl(n) = Ldln Jl(d)F(n/d). PROOF. o O = = f This shows O Remark. We have eonsidered eomplex-valued funetions on the positive integers. It is useful to notiee that Theorem 1 is valid whenever the funetions take their value in an abelian group. The proof goes through word for word.

Let Xo = x~c. Then axo == b (m). We have shown that ax == b (m) has a solution iff dlb. Suppose that Xo and Xt are solutions. axo == b (m) and aXt == b (m) imply that a(xl - xo) == O (m). Thus mla(xt - xo) and m'Ia'(xl - xo), which implies that m' Ix t - Xo or x t = Xo + km' for some integer k. One easily checks that any number of the form Xo + km' is a solution and that the solutions xo, Xo + m', ... , Xo + (d - 1)m' are inequivalent. Let Xl = Xo + km' be another solution. There are integers r and s such that k = rd + s and O ~ s < d.

11. Prove that 1k + 2k + ... + (p ţt(p - 1) - l)k == O (P) if p - Lrk and -1 (P) if p - 11k. 12. Use the existence of a primitive root to give another proof of Wilson's theorem (p - 1)! == -1 (p)' 13. Let G be a finite cyclic group and g E G a generator. Show that alI the other generators are of the form gk, where (k, n) = 1, n being the order of G. 14. Let A be a finite abelian group and a, bEA elements of order m and n, respectively. If (m, n) = 1, prove that ab has order mn. 15. Let K be a field and G s K* a finite subgroup of the multiplicative group of K.

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