Algebraic Number Theory

A Brief Introduction

J.S. Chahal

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000402254
• Edition: 1
• Topic: Mathématiques
• Subtopic: Théorie des nombres

This book offers the basics of algebraic number theory for students and others who need an introduction and do not have the time to wade through the voluminous textbooks available. It is suitable for an independent study or as a textbook for a first course on the topic.

The author presents the topic here by first offering a brief introduction to number theory and a review of the prerequisite material, then presents the basic theory of algebraic numbers. The treatment of the subject is classical but the newer approach discussed at the end provides a broader theory to include the arithmetic of algebraic curves over finite fields, and even suggests a theory for studying higher dimensional varieties over finite fields. It leads naturally to the Weil conjecture and some delicate questions in algebraic geometry.

About the Author

Dr. J. S. Chahal is a professor of mathematics at Brigham Young University. He received his Ph.D. from Johns Hopkins University and after spending a couple of years at the University of Wisconsin as a post doc, he joined Brigham Young University as an assistant professor and has been there ever since. He specializes and has published several papers in number theory. For hobbies, he likes to travel and hike. His book, Fundamentals of Linear Algebra, is also published by CRC Press.

1

Genesis: What Is Number Theory?

1.1 What Is Number Theory?

Number Theory is the study of numbers, in particular the whole numbers $1,2,3,\dots$, also called the natural numbers. The set of natural numbers is denoted by ℕ. Leaving aside the unit 1, these numbers fall into two categories: The indivisible numbers $2,3,5,7,\dots$ are the primes, and the rest $4,6,8,9,10,\dots$ composed of primes, are the composite numbers. The following basic facts, with proofs, about these numbers were already known to Euclid around 300 B.C.

Theorem 1.1. There are infinitely many primes.

Theorem 1.2 (Fundamental Theorem of Arithmetic). Every natural number $n>1$ is a unique product

$$\begin{array}{l}n=p_1^{e_1}\dots p_r^{e_r}(r\ge 1)\hfill \end{array}$$(1.1)

of powers of distinct primes $p_1,\dots ,p_r$ , taken in some order.

By looking at the list of primes, one can ask several naive but still unanswered questions. For example, is there an endless supply of twin primes? We call a pair of primes q, ptwin primes if $p=q+2$. [This is the closest two odd primes can be to each other.] A glance at the list

$$\begin{array}{l}3,5;5,7;11,13;17,19;29,31;\dots \hfill \end{array}$$

suggests that there are infinitely many pairs of twin primes, but no one has ever been able to prove this so far. Another big problem in number theory is the unproven conjecture of Goldbach, which asserts that every even number larger than 2 is a sum of two primes.

Many questions in number theory arise naturally in the study of geometry. The most fundamental fact in Euclidean geometry is the theorem of Pythagoras, which may be called the fundamental theorem of geometry. Actually, it was known to the Egyptians and Babylonians about two thousand years earlier, but they had no rigorous proof of it like Euclid did.

Theorem 1.3 (Fundamental Theorem of Geometry). The real numbers $0<x\le y<z$ are the side lengths of a right triangle if and only if

$$\begin{array}{l}{x}^2+y^2=z^2.\hfill \end{array}$$(1.2)

To number theorists, the most interesting solutions of (1.2), called the Pythagorean triples, are those with x, y, z whole numbers, such as $(3,4,5)$, $(5,12,13)$. It is said that the Egyptians used long ropes divided into three parts by two knots of lengths 3, 4 and 5 units...