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

By Kenneth Ireland, Michael Rosen

This well-developed, obtainable textual content information the old improvement of the topic all through. It additionally presents wide-ranging insurance of important effects with relatively straightforward proofs, a few of them new. This moment version comprises new chapters that supply a whole facts of the Mordel-Weil theorem for elliptic curves over the rational numbers and an outline of modern growth at the mathematics of elliptic curves.

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IfIX E Zero] and IX == 1 (A), prove that 1X 3 == 1 (9). ) 26. UseExercise25toshowthatif~,11,'EZ[ro]arenotzeroand~3 A divides at least one of the elements ~, 11, ,. + 11 3 + ,3 = O,then Chapter 4 The Structure of U(7L/n7L) Having introduced the notion of congruence and discussed some of its properties and applications we shall now go more deeply into the subject. The key result is the existence of primitive roots modulo a prime. This theorem was used by mathematicians before Gauss but he was the jirst to give a proof In the terminology introduced in Chapter 3 the existence of primitive roots is equivalent to the fact that U(7L/p7L) is a cyclic group when p is a prime.

Since pa/p = pa-l numbers less than pa are divisible by p, pa - pa- 1 numbers are prime to p. Notice that pa - pa-l = pa(l - lip). It folIows that (m) = m (1 - lip). We proved this formula in Chapter 2 in a different manner. 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 ofunits U(7L/m7L). To go more deeply into the structure of these algebraic objects we write m = p~l p~2 ...

An) and (b l , ... , bn) are called equivalent if ai == b i for i = 1, ... , n. The number ofsolutions tof(x l , ... , x n) == O (m) is defined to be the number of inequivalent solutions. For example, 3, 8, and 13 are solutions to 6x == 3 (15). 18 is also a solution, but the solution x = 18 is equivalent to the solution x = 3. It is useful to consider the matter from another point of view. The map from Z to Z/mZ given by a --+ a is a homomorphism. Iff(at, ... , an) == O(m), then f(at, ... , an) = O.

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