An Introduction to Abstract Algebra [2 ed.] 9783110198164

Table of contents :
1 Sets, relations and functions
1.1 Sets and subsets
1.2 Relations, equivalence relations and partial orders
1.3 Functions
1.4 Cardinality
2 The integers
2.1 Well-ordering and mathematical induction
2.2 Division in the integers
2.3 Congruences
3 Introduction to groups
3.1 Permutations of a set
3.2 Binary operations: semigroups, monoids and groups
3.3 Groups and subgroups
4 Cosets, quotient groups and homomorphisms
4.1 Cosets and Lagrange’s Theorem
4.2 Normal subgroups and quotient groups
4.3 Homomorphisms of groups
5 Groups acting on sets
5.1 Group actions and permutation representations
5.2 Orbits and stabilizers
5.3 Applications to the structure of groups
5.4 Applications to combinatorics – counting labellings and graphs
6 Introduction to rings
6.1 Definition and elementary properties of rings
6.2 Subrings and ideals
6.3 Integral domains, division rings and fields
7 Division in rings
7.1 Euclidean domains
7.2 Principal ideal domains
7.3 Unique factorization in integral domains
7.4 Roots of polynomials and splitting fields
8 Vector spaces
8.1 Vector spaces and subspaces
8.2 Linear independence, basis and dimension
8.3 Linear mappings
8.4 Orthogonality in vector spaces
9 The structure of groups
9.1 The Jordan-Holder Theorem
9.2 Solvable and nilpotent groups
9.3 Theorems on finite solvable groups
10 Introduction to the theory of fields
10.1 Field extensions
10.2 Constructions with ruler and compass
10.3 Finite fields
10.4 Applications to latin squares and Steiner triple systems
11 Galois theory
11.1 Normal and separable extensions
11.2 Automorphisms of field extensions
11.3 The Fundamental Theorem of Galois Theory
11.4 Solvability of equations by radicals
12 Further topics
12.1 Zorn’s Lemma and its applications
12.2 More on roots of polynomials
12.3 Generators and relations for groups
12.4 An introduction to error correcting codes
Bibliography
Index of notation
Index

Citation preview

de Gruyter Textbook Robinson · An Introduction to Abstract Algebra

Derek J. S. Robinson

An Introduction to Abstract Algebra

w DE

G

Walter de Gruyter Berlin · New York 2003

Author Derek J. S. Robinson Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, Illinois 61801-2975 USA

Mathematics Subject Classification 2000: 12-01, 13-01, 16-01, 20-01 Keywords: group, ring, field, vector space, Polya theory, Steiner system, error correcting code

© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress Cataloging-in-Publication

Data

Robinson, Derek John Scott. An introduction to abstract algebra / Derek J. S. Robinson. p. cm. - (De Gruyter textbook) Includes bibliographical references and index. ISBN 3-11-017544-4 (alk. paper) 1. Algebra, Abstract. I. Title. II. Series. QA162.R63 2003 512'.02—dc21 2002041471

ISBN 3 11 017544 4 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at . © Copyright 2003 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Cover design: Hansbernd Lindemann, Berlin. Typeset using the author's TgX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen.

In Memory of My Parents

Preface

The origins of algebra are usually traced back to Muhammad ben Musa al-Khwarizmi, who worked at the court of the Caliph al-Ma'mun in Baghdad in the early 9th Century. The word derives from the Arabic al-jabr, which refers to the process of adding the same quantity to both sides of an equation. The work of Arabic scholars was known in Italy by the 13th Century, and a lively school of algebraists arose there. Much of their work was concerned with the solution of polynomial equations. This preoccupation of mathematicians lasted until the beginning of the 19th Century, when the possibility of solving the general equation of the fifth degree in terms of radicals was finally disproved by Ruffini and Abel. This early work led to the introduction of some of the main structures of modern abstract algebra, groups, rings and fields. These structures have been intensively studied over the past two hundred years. For an interesting historical account of the origins of algebra the reader may consult the book by van der Waerden [15]. Until quite recently algebra was very much the domain of the pure mathematician; applications were few and far between. But all this has changed as a result of the rise of information technology, where the precision and power inherent in the language and concepts of algebra have proved to be invaluable. Today specialists in computer science and engineering, as well as physics and chemistry, routinely take courses in abstract algebra. The present work represents an attempt to meet the needs of both mathematicians and scientists who are interested in acquiring a basic knowledge of algebra and its applications. On the other hand, this is not a book on applied algebra, or discrete mathematics as it is often called nowadays. As to what is expected of the reader, a basic knowledge of matrices is assumed and also at least the level of maturity consistent with completion of three semesters of calculus. The object is to introduce the reader to the principal structures of modern algebra and to give an account of some of its more convincing applications. In particular there are sections on solution of equations by radicals, ruler and compass constructions, Polya counting theory, Steiner systems, orthogonal latin squares and error correcting codes. The book should be suitable for students in the third or fourth year of study at a North American university and in their second or third year at a university in the United Kingdom. There is more than enough material here for a two semester course in abstract algebra. If just one semester is available, Chapters 1 through 7 and Chapter 10 could be covered. The first two chapters contain some things that will be known to many readers and can be covered more quickly. In addition a good deal of the material in Chapter 8 will be familiar to anyone who has taken a course in linear algebra. A word about proofs is in order. Often students from outside mathematics question the need for rigorous proofs, although this is perhaps becoming less common. One

viii

Preface

answer is that the only way to be certain that a statement is correct or that a computer program will always deliver the correct answer is to prove it. As a rule complete proofs are given and they should be read, although on a first reading some of the more complex arguments could be omitted. The first two chapters, which contain much elementary material, are a good place for the reader to develop and polish theorem proving skills. Each section of the book is followed by a selection of problems, of varying degrees of difficulty. This book is based on courses given over many years at the University of Illinois at Urbana-Champaign, the National University of Singapore and the University of London. I am grateful to many colleagues for much good advice and lots of stimulating conversations: these have led to numerous improvements in the text. In particular I am most grateful to Otto Kegel for reading the entire text. However full credit for all errors and mis-statements belongs to me. Finally, I thank Manfred Karbe, Irene Zimmermann and the staff at Walter de Gruyter for their encouragement and unfailing courtesy and assistance. Urbana, Illinois, November 2002

Derek Robinson

Contents

1

Sets, 1.1 1.2 1.3 1.4

2

The 2.1 2.2 2.3

3

Introduction to groups 3.1 Permutations of a set 3.2 Binary operations: semigroups, monoids and groups 3.3 Groups and subgroups

31 31 39 44

4

Cosets, quotient groups and homomorphisms 4.1 Cosets and Lagrange's Theorem 4.2 Normal subgroups and quotient groups 4.3 Homomorphisms of groups

52 52 60 67

5

Groups acting on sets 5.1 Group actions and permutation representations 5.2 Orbits and stabilizers 5.3 Applications to the structure of groups 5.4 Applications to combinatorics - counting labellings and graphs . . . .

78 78 81 85 92

6

Introduction to rings 6.1 Definition and elementary properties of rings 6.2 Subrings and ideals 6.3 Integral domains, division rings and

7

relations and functions Sets and subsets Relations, equivalence relations and partial orders Functions Cardinality

1 1 4 9 13

integers Well-ordering and mathematical induction Division in the integers Congruences

Division in rings 7.1 Euclidean domains 7.2 Principal ideal domains 7.3 Unique factorization in integral domains 7.4 Roots of polynomials and splitting

17 17 19 24

fields

fields

99 99 103 107 115 115 118 121 127

χ

Contents

8

Vector spaces 8.1 Vector spaces and subspaces 8.2 Linear independence, basis and dimension 8.3 Linear mappings 8.4 Orthogonality in vector spaces

134 134 138 147 155

9

The 9.1 9.2 9.3

163 163 171 178

structure of groups The Jordan-Holder Theorem Solvable and nilpotent groups Theorems on finite solvable groups

10 Introduction to the theory of fields 10.1 Field extensions 10.2 Constructions with ruler and compass 10.3 Finite fields 10.4 Applications to latin squares and Steiner triple systems

185 185 190 195 199

11 Galois theory 11.1 Normal and separable extensions 11.2 Automorphisms of field extensions 11.3 The Fundamental Theorem of Galois Theory 11.4 Solvability of equations by radicals

208 208 213 221 228

12 Further topics 12.1 Zorn 's Lemma and its applications 12.2 More on roots of polynomials 12.3 Generators and relations for groups

235 235 240 243

12.4 An introduction to error correcting codes

254

Bibliography

267

Index of notation

269

Index

273

Chapter 1

Sets, relations and functions

The concepts introduced in this chapter are truly fundamental and underlie almost every branch of mathematics. Most of the material is quite elementary and will be familiar to many readers. Nevertheless readers are encouraged at least to review the material to check notation and definitions. Because of its nature the pace of this chapter is brisker than in subsequent chapters.

1.1

Sets and subsets

By a set we shall mean any well-defined collection of objects, which are called the elements of the set. Some care must be exercised in using the term "set" because of Bertrand Russell's famous paradox, which shows that not every collection can be regarded as a set. Russell considered the collection C of all sets which are not elements of themselves. If C is allowed to be a set, a contradiction arises when one inquires whether or not C is an element of itself. Now plainly there is something suspicious about the idea of a set being an element of itself, and we shall take this as evidence that the qualification "well-defined" needs to be taken seriously. A collection that is not a set is called a proper class. Sets will be denoted by capital letters and their elements by lower case letters. The standard notation a e A means that α is a element of the set A, (or a belongs to A). The negation of α e A is denoted by α £ A. Sets can be defined either by writing their elements out between braces, as in {a, b, c, d], or alternatively by giving a formal description of the elements, the general format being A = {a I a has property P}, i.e., A is the set of all objects with the property P. If A is a finite set, the number of its elements is written |A|. Subsets.

Let A and Β be sets. If every element of A is an element of B, we write Ac Β

2

1 Sets, relations and functions

and say that A is a subset of B, or that A is contained in B. If A ç Β and Β ç A, so that A and Β have exactly the same elements, then A and Β are said to be equal, A = B. The negation of this is A ^ Β. The notation A c Β is used if A ç Β and Α φ- B\ then A is a proper subset of B. Special sets. A set with no elements at all is called an empty set. An empty set E is a subset of any set A; for if this were false, there would be an element of E that is not in A, which is certainly wrong. As a consequence there is just one empty set; for if E and E ' are two empty sets, then E ç E' and E ' ç E, so that E = E'. This unique empty set is written 0.

Some further standard sets with a reserved notation are: N, Z, Q, M, C, which are respectively the sets of natural numbers 0 , 1 , 2 , . . . , integers, rational numbers, real numbers and complex numbers. Set operations. Next we recall the familiar set operations of union, intersection and complement. Let A and Β be sets. The union A U Β is the set of all objects which belong to A or Β (possibly both); the intersection Α Γι Β consists of all objects that belong to both A and Β. Thus AVJ Β = {χ \ χ e A or χ e B}, while ACiB

= {x\xeA

and χ e Β}.

It should be clear how to define the union and intersection of an arbitrary collection of sets {A^ I λ e Λ}; these are written | J Αλ

and

λεΛ

Q

Αλ.

λεΛ

The relative complement of Β in A is A-B

= {x\xeA

and χ φ Β}.

Frequently one has to deal only with subsets of some fixed set U, called the universal set. If A ç î/, then the complement of A in U is A = U - A.

1.1 Sets and subsets

3

Properties of set operations. We list for future reference the fundamental properties of union, intersection and complement. (1.1.1) Let A, B,C be sets. Then the following statements are valid: (i) A U Β = Β U A and Α Π Β = Β Π Α (commutative laws). (ii) (A U Β) U C = Λ U (Β U C) and (Α Π Β) Π C = Λ Π (Β Π C) laws).

(associative

(iii) Α Π (Β U C) = (Α Π Β) U (Α Π C) and A U (Β Π C) = (A U Β) Π (A U C) (distributive laws). (iv) A U A = A = A n A . (v) A U 0 = A, A η 0 = 0. (vi) A - ( υ λ ε Λ Sx) = η λ ε Λ ( Α - Βλ) and A(De Morgan's Laws)}

( f \ e A Βλ) = U i e A ( ¿ -

Βλ)

The easy proofs of these results are left to the reader as an exercise: hopefully most of these properties will be familiar. Set products. Let A\, A2,..., A„ be sets. By an η-tuple of elements from A\, A2, ..., An is to be understood a sequence of elements a\, aj, • • •, an with a¡ e A¿. The η-tuple is usually written (a 1, ai..... an) and the set of all //-tuples is denoted by Αι χ A2 χ · · · χ A n . This is the set product (or cartesian product) of A\, A 2 , . . . , An. For example Μ χ M is the set of coordinates of points in the plane. The following result is a basic counting tool. (1.1.2) / / Α ι , A2, . . . , An are finite sets, then IA1XA2X· · -xA„| = IA1HA2I... \An\. Proof. In forming an «-tuple (a\,a2,... ,an) we have |Ai| choices for a\, IA2I choices for Β exists. If a(ai) = a f a ) , then, applying a - 1 to each side, we arrive at αϊ = A by ß(b) = a. Then aß(b) = a (a) = b andccß = idß. Also β a {a) = ß(b) = a; since every α in A arises in this way, ßa = id^ and β = α - 1 . • The next result records some useful facts about inverses. (1.3.2) (i) If a : A ( a - 1 ) - 1 = a.

Β is a bijective function, then so is α - 1 : Β —• A, and indeed

(ii) If a : A Β and β : Β C are bijective functions, then β o a : A —> C is bijective, and (β o a ) - 1 = α - 1 ο β~λ. Proof, (i) The equations α ο α - 1 = idß and α - 1 o a = id^ tell us that a is the inverse of a - 1 . (ii) Check directly that a~] ο β~λ is the inverse οί β o a, using the associative law twice: thus (β o a) o ( c ^ 1 ο β - 1 ) = ((β ο α) ο « - ' ) ο β-1 = (β ο (α ο α-- 1 )) ο β - 1 = (β oidA) ο β-1 = β ο β-1 = i d c . Similarly ( α " 1 ο β-1) ο (β ο α) = idA. • Application to automata. As an illustration of how the language of sets and functions may be used to describe information systems we shall briefly consider automata. An automaton is a theoretical device which is a basic model of a digital computer. It consists of an input tape, an output tape and a "head", which is able to read symbols

12

1 Sets, relations and functions

on the input tape and print symbols on the output tape. At any instant the system is in one of a number of states. When the automaton reads a symbol on the input tape, it goes to another state and writes a symbol on the output tape.

To make this idea precise we define an automaton A to be a 5-tuple (/,

Ο,Ξ,ν,σ)

where I and O are the respective sets of input and output symbols, S is the set of states, ν : I χ S - » O is the output function and σ : I χ S S is the next state function. The automaton operates in the following manner. If it is in state s e S and an input symbol i e I is read, the automaton prints the symbol v(i. s) on the output tape and goes to state σ(ζ, s). Thus the mode of operation is determined by the three sets /, O, S and the two functions ν, σ. Exercises (1.3) 1. Which of the following functions are injective, surjective, bijective? (a) a : M -> Ζ where αÇt) = [χ] (= the largest integer < χ). (b) ce : M > 0 M where ce (.τ) = log(x). (Here ŒP° = {χ . r e i , .ι > 0}). (e) a : A χ Β —>· Β χ A where a ({a, b)) = (£>, a). 2. Prove that the composite of injective functions is injective, and the composite of surjective functions is surjective. 3. Let a : A —> Β be a function between finite sets. Show that if |Λ| > |Z?|, then a cannot be injective, and if | A \ < \ Β \, then a cannot be surjective. 3

4. Define a : M —> M by α (χ) = -^χργ. Prove that a is bijective. 5. Give an example of functions α, β on a set A such that αοβ = id^ but β ο α φ id^. 6. Let α : A —> Β be a injective function. Show that there is a surjective function β :Β A such that β ο α = id¿.

1.4 Cardinality

13

7. Let a : A 5 be a surjective function. Show that there is an injective function β :Β A such that α ο β = ids. 8. Describe a simplified version of an automaton with no output tape where the output is determined by the new state - this is called a state output automaton. 9. Let a : A Β be a function. Define a relation Ea on A by the rule: a Ea a' means that a (a) = a(a'). Prove that Ea is an equivalence relation on A. Then show that conversely, if E is any equivalence relation on a set A, then E = Ea for some function a with domain A.

1.4

Cardinality

If we want to compare two sets, a natural basis for comparison is the "size" of each set. If the sets are finite, their sizes are just the numbers of elements in the set. But how can one measure the size of an infinite set? A reasonable point of view would be to hold that two sets have the same size if their elements can be paired off. Certainly two finite sets have the same number of elements precisely when their elements can be paired. The point to notice is that this idea also applies to infinite sets, making it possible to give a rigorous definition of the size of a set, its cardinal. Let A and Β be two sets. Then A and Β are said to be equipollent if there is a bijection a : A - » B: thus the elements of A and Β may be paired off as (a, aia)), a e A. It follows from (1.3.2) that equipollence is an equivalence relation on the class of all sets. Thus each set A belongs to a unique equivalence class, which will be written

|A| and called the cardinal of A. Informally we think of |A| as the collection of all sets with the same "size" as A. A cardinal number is the cardinal of some set. If A is a finite set with exactly η elements, then A is equipollent to the set {0, 1 , . . . , η — 1} and |A| = |{0, 1, . . . , η — 1}|. It is reasonable to identify the finite cardinal |{0, 1, . . . , « — 1}| with the non-negative integer n. For then cardinal numbers appear as infinite versions of the non-negative integers. Let us sum up our very elementary conclusions so far. (1.4.1) (i) Every set A has a unique cardinal number | A|. (ii) Two sets are equipollent if and only if they have the same cardinal. (iii) The cardinal of a finite set is to be identified with the number of its elements. Since we plan to use cardinals to compare the sizes of sets, it makes sense to define a "less than or equal to" relation < on cardinals as follows: |A| < | 5 | means that there is an injective function α : A if |A| < \B\ and |A| φ | 5 | .

B. Of course we will write |A| < | 5 |

14

1 Sets, relations and functions

It is important to realize that this definition of < depends only on the cardinals |A| and \B\, not on the sets A and B. For if A' e |A| and B' e | 5 | , then there are bijections a' : A' -> A and β' : Β Β'; by composing these with a : A -> Β we obtain the injection β' ο α o a' : A' Β'. Thus | A'\ < \B'\. Next we prove a famous result about inequality of cardinals. (1.4.2) (The Cantor 4 -Bernstein 5 Theorem) If A and Β are sets such that |A| < | 5 | and\B\ < |A|, then |A| = | 5 | . The proof of (1.4.2) is our most challenging proof so far and some readers may prefer to skip it. However the basic idea behind it is not difficult. Proof. By hypothesis there are injective functions a \ A Β and β : Β -> Α. These will be used to construct a bijective function γ : A -> Β, which will show that |A| = | 5 | . Consider an arbitrary element a in A; then either a = ß(b) for some unique b e Β or else a ^ Im(ß): here we use the injectivity of β. Similarly, either b = a(a') for a unique a' e A or else b φ Im (α). Continuing this process, we can trace back the "ancestry" of the element a. There are three possible outcomes: (i) we eventually reach an element of A — Im(ß); (ii) we eventually reach an element of Β — Im(a); (iii) the process continues without end. Partition the set A into three subsets corresponding to possibilities (i), (ii), (iii), and call them AA, AB, Ασο respectively. In a similar fashion the set Β decomposes into three disjoint subsets Β A, BB, Boo; for example, if b e BA, we can trace b back to an element of A — Im (β). Now we are in a position to define the function γ : A B. First notice that the restriction of a to AA is a bijection from AA to BA, and the restriction of a to Aoo is a bijection from Aoo to Boo. Also, if χ e AB, there is a unique element x' e Β Β such that β (χ') = χ. Now define a (χ) γ{χ) = • a{x) x'

if X e AA if X e Aoo if X e AB.

Then γ is the desired bijection. Corollary (1.4.3) The relation < is a partial order on cardinal

• numbers.

For we have proved antisymmetry in (1.4.2), while reflexivity and transitivity are clearly true by definition. In fact one can do better since < is even a linear order. This is because of: 4

G e o r g Cantor (1845-1918).

5

F e l i x Bernstein (1878-1956).

1.4 Cardinality

15

(1.4.4) (The Law of Trichotomy) If A and Β are sets, then exactly one of the following must hold: |A| {a} sets up an injection from A to Ρ (A). Next assume that |A| = |P(A)|, so that there is a bijection a : A Ρ (A). Of course at this point we are looking for a contradiction. The trick is to consider the subset Β = {a \ a e Α, α φ a (a)} of A. Then Β e Ρ (A), so Β = a (a) for some a e A. Now either a e Β or α φ Β. If a e Β, then α φ a (a) = B; if α φ Β = a (a), then a e Β. This is our contradiction. • Countable sets. denoted by

The cardinal of the set of natural numbers Ν = {0, 1, 2 , . . . } is Ko

where Κ is the Hebrew letter aleph. A set A is said to be countable if |A| < Ko. Essentially this means that the elements of A can be "labelled" by attaching to each element a natural number as a label. An uncountable set cannot be so labelled. We need to explain what is meant by an infinite set for the next result to be meaningful. A set A will be called infinite if it has a subset that is equipollent with N, i.e., if < |A|. An infinite cardinal is the cardinal of an infinite set. (1.4.6) Ko is the smallest infinite cardinal. Proof. If A is an infinite set, then A has a subset Β such that Ko = | 5 | . Hence K0 w3 for all integers η > 10. 4. Prove that 2" > n 4 for all integers η > 17. 5. Prove by mathematical induction that 6 | η 3 — η for all integers η > 0. 6. Use the alternate form of mathematical induction to show that any η cents worth of postage where η > 12 can be made up using 4 cent and 5 cent stamps. [Hint: first verify the statement for η < 15].

2.2

Division in the integers

In this section we establish the basic properties of the integers that relate to division, notably the Division Algorithm, the existence of greatest common divisors and the Fundamental Theorem of Arithmetic.

20

2 The integers

Recall that if a, b are integers, then a divides b, in symbols a I b, if there is an integer c such that b = ac. The following properties of division are simple consequences of the definition, as the reader should verify. (2.2.1) (i) The relation of division is a partial order on Z. (ii) If a I b and a \ c, then a \ bx + cy for all integers χ, y. (iii) a I 0 for all a, while 0 | a only if a = 0. (iv) 1 I a for all a, while a \ 1 only if a = ± 1 . The Division Algorithm. The first result about the integers of real significance is the Division Algorithm·, it codefies the time-honored process of dividing one integer by another to obtain a quotient and remainder. It should be noted that the proof of the result uses WO. (2.2.2) Let a, b be integers with b φ 0. Then there exist unique integers q (the quotient) and r (the remainder) such that a = bq + r and 0 < r < \b\. Proof. Let S be the set of all non-negative integers of the form a — bq, where q e Ζ. In the first place we need to observe that S is not empty. For if b > 0, and we choose an integer q < then a — bq > 0; if b < 0, choose an integer q > so that again a—bq > 0 . Applying the Well-Ordering Law to the set S, we conclude that it contains a smallest element, say r. Then r = a — bq for some integer q, and a = bq + r. Now suppose that r > \b\. If b > 0, then a — b(q + 1 ) = r — b < r, while if b < 0, then a — b(q — 1) = r + b < r. In each case a contradiction is reached since we have produced an integer in S which is less than r. Hence r < \h\. Finally, we must show that q and r are unique. Suppose that a = bq' + r' where q', r' e Ζ and 0 0. • The integer d of (2.2.3) is called the greatest common divisor oîai,a2, symbols d = gcd{αϊ, «2, · · ·, a„j. If d = 1, the integers a¡, a2,...,

• • • ,an,in

an are said to be relatively prime ; of course this

means that these integers have no common divisors except ± 1 . The Euclidean 1 Algorithm. The proof of the existence of gcd's which has just been given is not constructive, i.e., it does not provide a method for calculating gcd's. Such a method is given by a well-known procedure called the Euclidean Algorithm. Assume that a, b are integers with b φ 0. Apply the Division Algorithm successively for division by b, r\, r 2 , . . . to get a = bq 1 + n>

0 < Π