Download A panorama in number theory, or, The view from Baker's by Gisbert Wüstholz PDF

By Gisbert Wüstholz

ISBN-10: 0521807999

ISBN-13: 9780521807999

Alan Baker's sixtieth birthday in August 1999 provided a great chance to arrange a convention at ETH Zurich with the objective of offering the state-of-the-art in quantity conception and geometry. the various leaders within the topic have been introduced jointly to offer an account of analysis within the final century in addition to speculations for attainable extra study. The papers during this quantity hide a wide spectrum of quantity conception together with geometric, algebrao-geometric and analytic facets. This quantity will attract quantity theorists, algebraic geometers, and geometers with a bunch theoretic historical past. even though, it's going to even be precious for mathematicians (in specific learn scholars) who're drawn to being expert within the kingdom of quantity idea first and foremost of the twenty first century and in attainable advancements for the long run.

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Extra info for A panorama in number theory, or, The view from Baker's garden

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Our first task, covered in Part 1, is to try to understand this concept of “degree” in as much detail as possible. This will lead to the developments we outlined in the Preface. PART I • • • • • DEGREE This page intentionally left blank Chapter 1 • • • • • DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two different definitions of the degree of an algebraic curve.

Arithmetic Modulo a Prime There is a special symbol used when working with the integers modulo p (where p is any prime): Fp . The letter “F” here stands for field. So Z/pZ and Fp are the same thing. Fp is defined as the set {0, 1, . . , p − 1} with addition and multiplication defined as follows: If x, y, and z are in Fp , x + y = z in Fp exactly when x + y ≡ z (mod p) and xy = z in Fp exactly when xy ≡ z (mod p). In other words, we allow ourselves to use equality signs in Fp where we would use congruence signs among integers.

During our journey of discovery, we will often use elliptic curves as typical examples of algebraic curves. Often, we’ll use y2 = x3 − x or y2 = x3 + 3x as our examples. 1. Greek Mathematics In this chapter, we will begin exploring the concept of the degree of an algebraic curve—that is, a curve that can be defined by polynomial equations. We will see that a circle has degree 2. The ancient Greeks also studied lines and planes, which have degree 1. Euclid limited himself to a straightedge and compass, which can create curves only of degrees 1 and 2.

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