Advanced Number Theory

Harvey Cohn

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486149240
• Topic: Matemáticas
• Subtopic: Teoría de números

`A very stimulating book ... in a class by itself.` — American MathematicalMonthly
Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.
The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker's Basis Theorem for Abelian Groups.
Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski's theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints.
In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory.

PART 1

BACKGROUND
MATERIAL

chapter I

Review of elementary number theory and group theory

NUMBER THEORETIC CONCEPTS

1. Congruence

We begin with the concept of divisibility. We say¹ a divides b if there is an integer c such that b = ac. If a divides b, we write a | b, and if a does not divide b we write a × b. If k ≥ 0 is an integer for which a^k | b but a^k+1 × b, we write a^k || b, which we read as “a^k divides b exactly.”

If m | (x − y), we write

and say that x is congruent to y modulo m. The quantity m is called the modulus, and all numbers congruent (or equivalent) to x (mod m) are said to constitute a congruence (or equivalence) class. Congruence classes are preserved under the rational integral operations, addition, subtraction, and multiplication; or, more generally, from the congruence (1) we have

where f(x) is any polynomial with integral coefficients.

2. Unique Factorization

It can be shown that any two integers a and b not both 0 have a greatest common divisor d(>0) such that if t | a and t | b then t | d, and conversely, if t is any integer (including d) that divides d, then t | a and t | b. We write d = gcd (a, b) or d = (a, b). It is more important that for any a and b there exist two integers x and y such that

If d = (a, b) = 1, we say a and b are relatively prime.

One procedure for finding such integers x, y is known as the Euclidean algorithm. (This algorithm is referred to in Chapter VI in another connection, but it is not used directly in this book.)

We make more frequent use of the division algorithm, on which the Euclidean algorithm is based: if a and b are two integers (b ≠ 0), there exists a quotient q and a remainder r such that

and, most important, a ≡ r(mod b) where

The congruence classes are accordingly called residue (remainder) classes.

From the foregoing procedure it follows that if (a, m) = 1 then an integer x exists such that (x, m) = 1 and ax ≡ b (mod m). From this it also follows that the symbol b/a (mod m) has integral meaning and may be written as x if (a, m) = 1.

An integer p greater than 1 is said to be a prime if it has no positive divisors except p and 1. The mo...