# A Classical Introduction to Modern Number Theory (2nd by Michael Rosen, Kenneth Ireland

By Michael Rosen, Kenneth Ireland

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This well-developed, obtainable textual content info the old improvement of the topic all through. It additionally offers wide-ranging assurance of vital effects with relatively hassle-free proofs, a few of them new. This moment version includes new chapters that offer an entire facts of the Mordel-Weil theorem for elliptic curves over the rational numbers and an outline of contemporary growth at the mathematics of elliptic curves.

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Extra info for A Classical Introduction to Modern Number Theory (2nd Edition) (Graduate Texts in Mathematics, Volume 84)

Example text

2' has primitive rootsfor 1= 1 or 2 but notfor I ~ 3. If I ~ 3, then {( - 1t5 b Ia = 0, 1 and 0 ::; b < 2'- 2} constitutes a reduced residue system mod 2'. ) is the direct product of two cyclic groups, one oforder 2, the other of order 2'- 2. PROOF. 1 is a primitive root mod 2, and 3 is a primitive root mod 4. From now on let us assume that I ~ 3. We claim that (1) 5 2 ' - 3 == 1 + 2' - 1 (2 ') . This is true for 1= 3. Assume th at it is true for I ~ 3 and we shall prove it is true for I + 1. First notice that (1 + 2'-1)2 = 1 + 2' + 2 21- 2 and that 21 - 2 ~ I + 1 for I ~ 3.

0 Theorem 1. ) is a cyclic group. PROOF. ) of order d. ) satisfying x d == I form a group of order d. Thus Lcld rjJ(c) = d. Applying the Mobius inversion theorem we obtain rjJ(d) = Lid Il(c)d /c. 5. In particular, rjJ(p - I) = ¢(p - I), which is greater than I if p> 2. Since the case p = 2 is trivial, we have shown in all cases the existence of an element [in fact, ¢(p - I) elements] of order p - 1. 0 Theorem 1 is of fundamental importance. It was first proved by Gauss. After giving some new terminology we shall outline two more proofs.

Notice that p" - pa- 1 = pa(l - l ip). It follows that ¢(m) = m (1 - lip). We proved this formula in Chapter 2 in a different manner. r:'. n 36 3 Congruence Let us summarize. In treating a number of arithmetical questions, the notion of congruence is extremely useful. This notion led us to consider the ring 7L/m7L and its group of units U(7L/m7L). To go more deeply into the structure of these algebraic objects we write m = p~'p~' . . p~' and are led, via the Chinese Remainder Theorem, to the folIowing isomorphisms: 7L/m7L :;:: U(7L/m7L) :;:: 7L/p~'7L Ef> 7L/P'2'7L ED· ..