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#### ADVANCED NUMBER THEORY

**By Harvey Cohn**

**Dover Publications, Inc.**

**Copyright © 1962 Harvey Cohn**

All rights reserved.

ISBN: 978-0-486-14924-0

All rights reserved.

ISBN: 978-0-486-14924-0

#### Contents

INTRODUCTORY SURVEY,PART 1. BACKGROUND MATERIAL,

1. Review of Elementary Number Theory and Group Theory,

2. Characters,

3. Some Algebraic Concepts,

4. Basis Theorems,

5. Further Applications of Basis Theorems,

PART 2. IDEAL THEORY IN QUADRATIC FIELDS,

6. Unique Factorization and Units,

7. Unique Factorization into Ideals,

8. Norms and Ideal Classes,

9. Class Structure in Quadratic Fields,

PART 3. APPLICATIONS OF IDEAL THEORY,

10. Class Number Formulas and Primes in Arithmetic Progression,

11. Quadratic Reciprocity,

12. Quadratic Forms and Ideals,

13. Compositions, Orders, and Genera,

CONCLUDING SURVEY,

Bibliography and Comments,

Appendix Tables,

Index,

CHAPTER 1

**Review of elementary number theory and group theory**

NUMBER THEORETIC CONCEPTS

**1. Congruence**

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 *x*4 + *y*4 = *x*7*y* + 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: *r*1 when *x* is divided by 3, *r*2 when *x* is divided by 4, and *r*3 when *x* is divided by 5. Then

*x* [equivalent to] 40*r*1 + 45*r*2 + 36*r*3 (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) *b*φ(*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)

by writing

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 *p*2, 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 *p*2). 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]

*(Continues...)*

Excerpted fromADVANCED NUMBER THEORYbyHarvey Cohn. Copyright © 1962 Harvey Cohn. Excerpted by permission of Dover Publications, Inc..

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