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"A very stimulating book ... in a class by itself." — American Mathematical Monthly
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, ...
"A very stimulating book ... in a class by itself." — American Mathematical Monthly
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.
Review of elementary number theory and group theory
NUMBER THEORETIC CONCEPTS
We begin with the concept of divisibility. We say 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 ak | b but ak+1 × b, we write ak || b, which we read as "ak divides b exactly."
If m | (x - y), we write
(1) x [equivalent to] y(mod m)
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
(2) f(x) [equivalent to] f(y) (mod m)
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
(1) ax + by = d.
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
(2) a = qb + r
and, most important, a [equivalent to] r(mod b) where
(3) 0 ≤ r< |b|.
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 [equivalent to] 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 most important result of the Euclidean algorithm is the theorem that if the prime p is such that p | ab then p | a or p | b. Thus, by an elementary proof, any nonzero integer m is representable in the form
(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where the pi are distinct primes. The representation is unique within rearrangement of factors. Each factor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called primary.
EXERCISE 1. Observe that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Write down and prove a general theorem enabling us to use ordinary arithmetic to work with fractions modulo m (if the denominators are prime to m).
EXERCISE 2. Prove [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
EXERCISE 3. From the remarkable coincidence 24 + 54 = 27 · 5 + 1 = 641 show 232 + 1 [equivalent to] 0 (mod 641). Hint. Eliminate y between the pair of equations x4 + y4 = x7y + 1 = 0 and carry the operations over to integers (mod 641).
EXERCISE 4. Write down and prove the theorem for the solvability or nonsolvability of ax [equivalent to] b (mod m) when (a, m) > 1.
3. The Chinese Remainder Theorem
If m = rs where r > 0, s > 0, then every congruence class modulo m corresponds to a unique pair of classes in a simple way, i.e., if x [equivalent to] y (mod m), then x [equivalent to] y (mod r) and x [equivalent to] y (mod s). If (r, s) = 1, the converse is also true; every pair of residue (congruence) classes modulo r and modulo s corresponds to a single residue class modulo rs. This is called the Chinese remainder theorem. One procedure for defining an x such that x [equivalent to] a (mod r) and x [equivalent to] b (mod s) uses the Euclidean algorithm, since (x =)a + rt = b + su constitutes an equation in the unknowns t and u, as in (1) of §2.
As a result of this theorem, if we want to solve the equation
(1) f(x) [equivalent to] 0 (mod m),
all we need do is factor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and then solve each of the equations
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
for as many roots as occur (possibly none). If xi is a solution to (2), we apply the Chinese remainder theorem step-by-step to solve simultaneously the equations
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
to obtain a solution to (1). If ri is the number of incongruent solutions to (2), there will be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] incongruent solutions. (The result is true even if one or more ri = 0.)
EXERCISE 5. In a game for guessing a person's age x, one discreetly requests three remainders: r1 when x is divided by 3, r2 when x is divided by 4, and r3 when x is divided by 5. Then
x [equivalent to] 40r1 + 45r2 + 36r3 (mod 60).
Discuss the process for the determination of the integers 40, 45, 36.
4. Structure of Reduced Residue Classes
A residue class modulo m will be called a reduced residue class (mod m) if each of its members is relatively prime to m. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (prime factorization), then any number x relatively prime to m may be determined modulo m by equations of the form
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The number of reduced residue classes modulo pa is given by the Euler function:
(2) φ(pa) = pa(1 - 1/p).
By the Chinese remainder theorem the number of reduced residue classes modulo m is (m), where
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
By the Fermat-Euler theorem, if (b, m) = 1, then
(4) bφ(m) [equivalent to] 1 (mod m).
A number g is a primitive root of m if
(5) gk [equivalent to] 1 (mod m) for 0 < k< φ(m).
Only the numbers m = pa, 2pa, 2, and 4 have primitive roots (where p is an odd prime). But then, for such a value of m, all y relatively prime to p are representable as
(6) y [equivalent to] gi(mod m),
where t takes on all (m) values; t = 0, 1, 2, ..., φ(m) - 1.
The accompanying tables (see appendix) give the minimum primitive root g for such prime p< 100 and represent y in terms of t and t in terms of y modulo p. Generally, t is called the index (abbr. I in the tables) and y is the number (abbr. N). Of course, the index is a value modulo φ(m), and the operation of the index recalls to mind elementary logarithms.
EXERCISE 6. Verify the index table modulo 19 and solve
210y60 [equivalent to] 1470 (mod 19)
10 ind 2 + 60 ind y ? 70 ind 14 (mod 18)
(and using Exercise 4, etc.).
5. Residue Classes for Prime Powers
In the case of an odd prime power pa, for a fixed base p, a single value g can be found that will serve as a primitive root for all exponents a > 1. In fact, g need be selected to serve only as the primitive root of p2, or, even more simply, as shown in elementary texts, g can be any primitive root of p with just the further property gp-1 [??] 1 (mod p2). We then take (6) of §4 to represent an arbitrary reduced residue class y(mod pa), using the minimum positive g for definiteness.
In the case of powers of 2, the situation is much more complicated. The easy results are (taking odd y) for different powers of 2
(1) y = 1 (mod 2), trivially,
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];
but for odd y, modulo 8, we find there is no primitive root. Thus there is no way of writing all odd y [equivalent to] gt (mod 8) for t = 0, 1, 2, 3. We must write
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
yielding the following table of all odd y modulo 8.
More generally, if we consider residues modulo 2a, a ≥ 3, we find the odd y are accounted for by
(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Excerpted from ADVANCED NUMBER THEORY by Harvey Cohn. Copyright © 1962 Harvey Cohn. Excerpted by permission of Dover Publications, Inc..
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