An Introduction to Abstract Algebra via Applications - New Mexico

151 Pages·2007·0.81 MB·English

An Introduction to Abstract Algebra
via Applications
David R. Finston and Patrick J. Morandi
Department of Mathematical Sciences
New Mexico State University
Las Cruces NM 88003-8001
September 25, 2007
ii
Contents
1 Identi(cid:133)cation Numbers and Modular Arithmetic 1
1.1 Examples of Identi(cid:133)cation Numbers . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Error Detection with Identi(cid:133)cation Numbers . . . . . . . . . . . . . . . . . . 19
2 Error Correcting Codes 23
2.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 The Hamming Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Coset Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5 The Golay Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Rings and Fields 43
3.1 The De(cid:133)nition of a Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 First Properties of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Linear Algebra and Linear Codes 59
4.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Linear Independence, Spanning, and Bases . . . . . . . . . . . . . . . . . . . 64
4.3 Linear Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Field Extensions and Ruler and Compass Constructions 75
5.1 Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Ruler and Compass Constructions . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Constructions and Field Extensions . . . . . . . . . . . . . . . . . . . . . . . 80
5.4 Classical Construction Problems . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.1 Angle Trisection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.2 Duplicating a Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.3 Squaring the Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4.4 Constructible Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . 86
iii
iv CONTENTS
6 Quotient Rings and Field Extensions 89
6.1 Arithmetic of Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Ideals and Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7 Cyclic Codes 103
7.1 Introduction to Cyclic Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3 Reed-Solomon Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.4 Error Correction for Reed-Solomon Codes . . . . . . . . . . . . . . . . . . . 116
8 Cryptography and Group Theory 121
8.1 The RSA encryption system . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.3 Secure Signatures with RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9 Symmetry 133
9.1 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.2 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
9.3 Examples of Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.4 Symmetry Groups of Bounded Figures . . . . . . . . . . . . . . . . . . . . . 140
Bibliography 143
List of Symbols 145
Index 145
Chapter 1
Identi(cid:133)cation Numbers and Modular
Arithmetic
The (cid:133)rst topic we will investigate in this course is the mathematics of identi(cid:133)cation numbers.
Many things are described by a code of digits; zip codes, items in a grocery store, and books,
to name three. One feature to all of these codes is the inclusion of an extra numerical digit,
called a check digit, designed to detect errors in reading the code. When a machine (or
a human) reads information, there is always the possibility of the information being read
incorrectly. For example, moisture or dirt on the scanner used by a grocery store clerk can
prevent an item(cid:146)s code from being read correctly. It would be unacceptable if, because of a
scanning error, a customer is charged for caviar when they are buying tuna (cid:133)sh. The use of
the check digit allows for the detection of some scanning errors. If an error is detected, the
item is then rescanned until the correct code is read.
1.1 Examples of Identi(cid:133)cation Numbers
There are many di⁄erent methods being used to produce identi(cid:133)cation numbers. We will
discuss three of them; the United States Postal Service zip code, the Universal Product Code
used for consumer products, and the International Standard Book Number.
The USPS Zip Code
The United States Postal Service uses a bar code to read zip codes on mail. The following
bar code is that for the Mathematical Sciences Department of NMSU, whose zip code is
88003-8001.
1
2 CHAPTER 1. IDENTIFICATION NUMBERS AND MODULAR ARITHMETIC
The bar code represents a ten digit number. There are (cid:133)fty two lines in the bar code. The
(cid:133)rst and last lines are just markers. The remaining (cid:133)fty lines comprise ten groups of (cid:133)ve,
and each group of (cid:133)ve represents a digit. The (cid:133)rst nine digits form the nine digit zip code
of the addressee. The tenth digit is a check digit. This digit is computed as follows: the
digits forming the zip code are added, and the check digit is the smallest nonnegative integer
needed to make the sum be divisible by 10. For example, given the zip code 88003-8001 for
the Department of Mathematical Sciences at New Mexico State University, the nine digits
sum to 28. Therefore, the check digit must be 2. Thus, the bar code this zip code represents
the ten digit number 8800380012.
This scheme allows one to determine the check digit for any nine digit zip code. For
example, if we only knew the nine digit zip code 88003-8001, the check digit x would be the
number between 0 and 9 such that the sum
8+8+0+0+3+8+0+0+1+x
was evenly divisible by 10. Since this sum is 28+x, the only choice for x is to be 2.
The purpose of the check digit is to detect errors in reading the code. For example,
suppose that the zip code 8800380012 was incorrectly read as 8800880012 by reading the
(cid:133)fth digit as an 8 instead of as a 3. The sumof the digits would then be 8+8+8+8+1+2 =
35, which is not divisible by 10. Therefore, the postal service(cid:146)s scanners would detect an
error, and the zip code would have to be read again.
The Universal Product Code (UPC)
The Universal Product Code, or UPC, appears on grocery items.
This is a twelve digit code consisting of two blocks of (cid:133)ve digits preceded and followed with
a single digit, as the example above indicates. The (cid:133)rst six identify the country and the
1.1. EXAMPLES OF IDENTIFICATION NUMBERS 3
manufacturer of the product and the next (cid:133)ve identify the product itself. The (cid:133)nal digit is
the check digit. A twelve digit code (a ;:::;a ) is valid provided that
1 12
3a +a +3a +a + +3a +a
1 2 3 4 11 12
(cid:1)(cid:1)(cid:1)
is evenly divisible by 10. The UPC of the example above is 0 41390 30860 4. Therefore, the
sum for this code is
3 0+1 4+3 1+1 3+3 9+1 0+3 3+1 0+3 8+1 6 +3 0+1 4 = 80:
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)
This sum is indeed evenly divisible by 10, so the number is valid.
As with the zip code, given the (cid:133)rst eleven digits, there is enough information to uniquely
determine the check digit. For example, given the partial UPC of 0 71142 00001, if the check
digit is x, then
3 0+7+3 1+1+3 4+2+3 0+0+3 0+0+3 1+x = 28+x;
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)
which forces x = 2.
As with the zip code scheme, the UPC adds the check digit to help detect errors. For
example, if the code 0 71142 00001 2 was incorrectly read as 0 71342 00001 2 by reading the
fourth digit as a 3 instead of as a 1, then the computation to check if this number is valid
would give
3 0+7+3 1+3+3 4+2+3 0+0+3 0++0+3 1+2 = 32;
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)
which is not divisible by 10. Therefore, a grocery store scanner would not recognize the code
as valid, and the cashier would have to rescan the item.
The International Standard Book Number (ISBN)
Books are identi(cid:133)ed by a ten digit number, abbreviated by ISBN. For example, the book
Field and Galois Theory, published by Springer, has ISBN 0-387-94753-1. The (cid:133)rst digit
identi(cid:133)es the language in which the book is written, the second block of digits identi(cid:133)es the
publisher, the third block identi(cid:133)es the book itself, and the (cid:133)nal digit is the check digit. In
this scheme, each digit can be a numeral 0;:::;9 or X, which represents 10. A ten digit
number (a ;:::;a ) is a valid ISBN provided that
1 10
10a +9a +8a + +2a +a
1 2 3 9 10
(cid:1)(cid:1)(cid:1)
is evenly divisible by 11. In the number above, we have
10 0+9 3+8 8+7 7+6 9+5 4+4 7+3 5+2 3+1
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)
= 11 24;
(cid:1)
4 CHAPTER 1. IDENTIFICATION NUMBERS AND MODULAR ARITHMETIC
so the number is indeed valid. The digit X is only used, when appropriate, for the check
digit.
As with the previous two examples, the check digit can be determined uniquely, given
that it is between 0 and 10. For example, for the book A Classical Introduction to Modern
Number Theory, published by Springer, whose number will start with 0-387-97329, the check
digit x must result in
10 0+9 3+8 8+7 7+6 9+5 7+4 3+3 2+2 9+x
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)
= 265+x divisible by 11.
Since 11 24 = 264, if 265+x is to be divisible by 11 and 0 x 10, then x = 10. Thus,
(cid:1) (cid:20) (cid:20)
the check digit for this book is X, and so the ISBN is 0-387-97329-X.
The ISBN scheme also allows for detection of some errors. When we discuss error de-
tection in more detail below, we will see that all of these schemes will detect an error in a
single digit. Unfortunately, errors in more than one digit are not always detected. However,
the ISBN scheme does better, in some sense, than the other two schemes above because it
detects transposition errors. For example, given the ISBN 0-387-97329-X, if the (cid:133)fth and
sixth digits are transposed, the resulting number is 0-387-79329-X. The check for validity of
this number would result in the sum
10 0+9 3+8 8+7 7+6 7+5 9+4 3+3 2+2 9+10 = 273;
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)
which is not divisible by 11. Thus, this number is invalid. However, transposing digits in a
valid zip code will always result in a number considered valid, since the sum of the digits is
unchanged by this transposition. We will discuss transposition errors in more detail later.
Exercises
1. Check if the following numbers are valid ISBNs:
(a) 0-8218-2169-5
(b) 0-201-01361-9
(c) 2-87647-089-6
(d) 3-7643-3065-1
2. Suppose a UPC is read, but the third digit is left out, and the result is 0 7x172 38175
1, where x represents the missing digit. Calculate, in terms of x, the sum needed to
check if this is a valid number. Then write down the condition on x required for the
number to be valid, and determine x.
3. The number 0-8176-3165-1 is an invalid ISBN (check this!). It was created by taking
the ISBN number of a book and changing one digit. Can you tell which digit was
changed? Explain why not by giving two examples of a valid ISBN that di⁄ers from
this one in exactly one digit.
1.2. MODULAR ARITHMETIC 5
4. Consider the following identi(cid:133)cation number scheme: If a = (a ;a ;a ;a ), where each
1 2 3 4
a is between 0 and 4, then the number a is valid provided that 4a +3a +a +2a
i 1 2 3 4
is divisible by 5. If (3;2;4;x) is a valid number, determine x.
5. Consider the following identi(cid:133)cation number scheme: a valid number is a 5-tuple of
integers a = (a ;a ;a ;a ;a ) with 0 a 12 such that 2a +3a +5a +a +6a is
1 2 3 4 5 i 1 2 3 4 5
(cid:20) (cid:20)
divisible by 13. If (2;3;4;11;x) is a valid number, determine x.
6. Consider the scheme of the previous problem. If (a ;a ;a ;a ;a ) is a valid number
1 2 3 4 5
and if a = a , prove that (a ;a ;a ;a ;a ) is not valid.
1 2 2 1 3 4 5
6
7. Let n be a positive integer and let m be a positive divisor of n. If a and b are integers
with a bmodn, prove that a bmodm.
(cid:17) (cid:17)
1.2 Modular Arithmetic
In order to investigate the error detection capabilities of the various identi(cid:133)cation number
schemes we have discussed, and to work with the other applications in this course, we will
look carefully at the computations involved in these schemes. In all three, a number is valid
if some combination of its entries is divisible by some speci(cid:133)c positive integer (10 or 11
in the examples). This actual result of the computation is not important in its own right.
Ratherwhat is importantis onlywhethertheresultis divisiblebythegiveninteger. Phrased
another way, what is important is not the combination but rather the remainder we would
get if we divide our speci(cid:133)c integer into the result. In some sense we are doing arithmetic
with these remainders when we do calculations in these schemes.
Consider the following well known scenario. When we tell time in the U.S., the hour
value is any whole number between 1 and 12. Three hours after 10 o(cid:146)clock will be 1 o(cid:146)clock.
In general, to see what time it will be n hours after 10 o(cid:146)clock, you add n to 10, and then
remove enough multiples of 12 until you have a value between 1 and 12. For instance, in
37 hours past 10 o(cid:146)clock, the time will be 11 o(cid:146)clock since 47 = 36+11. In telling time, we
then identify 13 o(cid:146)clock with 1 o(cid:146)clock, 14 o(cid:146)clock with 2 o(cid:146)clock, and so on. In this clock
arithmetic, if we add 12 hours to any time, we get the same time (but changing AM to PM
and vice-versa). Therefore, 12 acts in clock arithmetic like 0 acts in ordinary arithmetic.
There is nothing special about 12 with respect to obtaining a new type of arithmetic. As
we will see in more detail below, in doing calculations in the various identi(cid:133)cation number
schemes we talked about above, we are essentially doing clock arithmetic, but with 12 re-
placed by 10 for the zip code and UPC, and by 11 for the ISBN scheme. When we discuss
coding theory, we will use clock arithmetic with 12 replaced by 2, and when we discuss cryp-
tography, we will replace 12 by very large integers. We therefore need to discuss the general
notion of clock arithmetic.
We begin with a very familiar concept.
6 CHAPTER 1. IDENTIFICATION NUMBERS AND MODULAR ARITHMETIC
De(cid:133)nition 1.1. Let a and n be integers. We say that n divides a (or a is divisible by n) if
a = nb for some integer b:
De(cid:133)nition 1.2. Let n be a positive integer. We say that two integers a and b are congruent
modulo n if b a is divisible by n. When this occurs, we write a bmodn.
(cid:0) (cid:17)
Since b a is divisible by n exactly when b a = qn for some integer q, we see that
(cid:0) (cid:0)
a bmodn if b = a+qn for some q. This is a convenient way to express congruence modulo
(cid:17)
n in terms of an equation. If n = 12, then to say a bmod12 is equivalent to saying a
(cid:17)
o(cid:146)clock is the same time as b o(cid:146)clock, if we ignore AM and PM. Congruence modulo n is
a relation on the set of integers. The (cid:133)rst thing we point out is that this relation is an
equivalence relation.
Proposition 1.3. The relation congruence modulo n is an equivalence relation for any
positive integer n.
Proof. Let n be a positive integer. We must prove that congruence modulo n is re(cid:135)exive,
symmetric, and transitive. For re(cid:135)exivity, let a be any integer. Then a amodn since
(cid:17)
a a = 0 is divisible by n; for 0 = n 0. Next, for symmetry, suppose that a and b are
(cid:0) (cid:1)
integers with a bmodn. Then b a is divisible by n; say b a = qn for some integer
(cid:17) (cid:0) (cid:0)
n. Then a b = ( q)n, so a b is also divisible by n. Therefore, b amodn, and so this
(cid:0) (cid:0) (cid:0) (cid:17)
relation is symmetric. Finally, to prove transitivity, suppose that a;b;c are integers with
a bmodn and b cmodn. Then b a and c b are both divisible by n. Then b a = sn
(cid:17) (cid:17) (cid:0) (cid:0) (cid:0)
and c b = tn for some integers s;t. Adding these equations gives c a = (s+t)n, so c a
(cid:0) (cid:0) (cid:0)
is divisible by n, and so a cmodn. This proves transitivity. Since we have shown that
(cid:17)
congruence modulon is re(cid:135)exive, symmetric, andtransitive, it is anequivalence relation.
Understanding the equivalence classes of this relation is of crucial importance. Recall
that if is an equivalence relation on a set X, then the equivalence class of an element
(cid:24)
a X is the set . For ease of notation, in dealing with congruence modulo n, we shall write
2
the equivalence class of an integer a by a. Therefore, b X : b a
f 2 (cid:24) g
a = b Z : b amodn :
f 2 (cid:17) g
Suppose n = 12. The equivalence class of 1 consists of all integers that are congruent
to 1 modulo 12. That is, the equivalence class contains all integers c with c o(cid:146)clock equal to 1
o(cid:146)clock. Wehave1 = :::; 23; 11;1;13;25;::: . Similarly,2 = :::; 22; 10;2;14;26;::: .
f (cid:0) (cid:0) g f (cid:0) (cid:0) g
Note that 12 = :::; 12;0;12;::: contains 0. An equivalence class can be represented in
f (cid:0) g
di⁄erent ways. We have 12 = 0 = 24, and, more generally, 12 = 12n for any integer n. In
(cid:0)
other words, 12 is the equivalence class of any element of the set 12 = :::; 12;0;12;::: .
f (cid:0) g
If n is any positive integer, we denote by Zn the set of equivalence classes of integers for
the equivalence relation of congruence modulo n. For n = 2 we have
Z2 = 0;1
f g