<<

. 13
( 13)



2 4 8 8 8
x + x + x + x and 1.
4 3 2
9.3.
3 ’ i and 4 + 2i, or 4 ’ i and 1 ’ 2i, or 4 ’ 2i and ’3 + i.
9.5.
gcd(a, b) = 3, s = ’5, t = 4.
9.7.
gcd(a, b) = 1, s = ’(2x + 1)/3, t = (2x + 2)/3.
9.9.
gcd(a, b) = 2x + 1, s = 1, t = 2x + 1.
9.11.
gcd(a, b) = 1, s = 1, t = ’1 + 2i.
9.13.
x = ’6, y = 5. 9.17. x = ’14, y = 5.
9.15.
9.19. [23]. 9.21. [17].
9.25. (x ’ 1)(x 4 + x 3 + x 2 + x + 1).
9.23. No solutions.
(x 2 + 2)(x 2 + 3). 9.29. x 4 ’ 9x + 3.
9.27.
x 3 ’ √ + 1. √
9.31. 4x √ √
(x ’ 2)(x + 2)(x ’ i 2)(x + i 2)(x ’ 1 ’ i)(x ’ 1 + i)
9.33.
(x + 1 ’ i)(x + 1 + i).
(x 2 ’ 2)(x 2 + 2)(x 2 ’ 2x + 2)(x 2 + 2x + 2).
9.35.
x 5 + x 3 + 1, x 5 + x 2 + 1, x 5 + x 4 + x 3 + x 2 + 1, x 5 + x 4 + x 3 + x + 1,
9.37.
x 5 + x 4 + x 2 + x + 1, x 5 + x 3 + x 2 + x + 1.
x 3 + 2.
9.39.

Kerψ = {q(x) · (x 2 ’ 2x + 4)|q(x) ∈ Q[x]} and Im ψ = Q( 3i) =
9.41. √
{a + b 3i|a, b ∈ Q}.
9.43. Irreducible by Eisenstein™s Criterion.
9.45. Irreducible, since it has no linear factors.
Reducible; any polynomial of degree >2 in R[x] is reducible.
9.47.
9.49. No. 9.55. No.
x ≡ 40 mod 42. 9.63. x ≡ 22 mod 30.
9.61.
9.67. 65.


CHAPTER 10

10.1. ((0, 0)), ((0, 1)), ((1, 0)), Z2 — Z2 .
10.3. (0) and Q.
316 ANSWERS TO THE ODD-NUMBERED EXERCISES


10.5. (p(x)) where p(x) ∈ C[x].
10.7. The quotient ring is a ¬eld.


+ (3) + 1 (3) + 2
(3)
(3) + 1 (3) + 2
(3) (3)
(3) + 1 (3) + 1 (3) + 2 (3)
(3) + 2 (3) + 2 (3) + 1
(3)

· (3) + 1 (3) + 2
(3)
(3) (3) (3) (3)
(3) + 1 (3) + 1 (3) + 2
(3)
(3) + 2 (3) + 2 (3) + 1
(3)

10.9. The ideal ((1, 2)) is the whole ring Z3 — Z3 . The quotient ring is not a ¬eld.


·
+ ((1, 2))
((1, 2))
((1, 2)) ((1, 2))
((1, 2)) ((1, 2))

10.11. 8x + 2 and 14x + 97. 10.13. x 2 + x and x 2 .
10.17. (a) 6; (b) 36; (c) x 2 ’ 1, (a) © (b) = (lcm(a, b)).
10.33. No. 10.35. The whole ring.
Z8
10.37.
|
([2]8 )
|
([4]8 )
|
([0]8 )

Irreducible; Z11√
10.39. . 10.41. Reducible. √√
Irreducible; Q( 2). 10.45. Irreducible; Q( 2, 3).
4
10.43.
10.47. Not a ¬eld; contains zero divisors.
10.49. A ¬eld by Corollary 10.16. 10.51. A ¬eld by Theorem 10.17.
Not a ¬eld; x + 1 = (x + 2)(x + 3) in Z5 [x].
2
10.53.
A ¬eld isomorphic to Q[x]/(x 4 ’ 11).
10.55.
(0) and (x n ) for n 0; (x) is maximal.
10.59.
ANSWERS TO THE ODD-NUMBERED EXERCISES 317


CHAPTER 11

11.1. GF(5) = Z5 = {0, 1, 2, 3, 4}.

+ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 0
2 2 3 4 0 1
3 3 4 0 1 2
4 4 0 1 2 3

· 0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 1 3
3 0 3 1 4 2
4 0 4 3 2 1

11.3. GF(9) = Z3 [x]/(x 2 + 1) = {a± + b|a, b ∈ Z3 , ± 2 + 1 = 0}.

+ ± +1 ± +2 2± + 1 2± + 2
0 1 2 2±
±

±+1 ±+2 2± + 1 2± + 2
0 0 1 2 2±
±
±+1 ±+2 2± + 1 2± + 2
1 1 2 0 2±
±
±+2 ±+1 2± + 2 2± + 1
2 2 0 1 2±
±
± +1 ±+2 2± + 1 2± + 2
2± 0 1 2
± ±
±+1 ±+1 ± +2 2± + 1 2± + 2 2± 1 2 0
±
±+2 ±+2 ±+1 2± + 2 2± + 1
2± 2 0 1
±
2± + 1 2± + 2 ±+1 ±+2
2± 2± 0 1 2 ±
2± + 1 2± + 1 2± + 2 ±+1 ±+2
2± 1 2 0 ±
2± + 2 2± + 2 2± + 1 ±+2 ±+1
2± 2 0 1 ±

· ± +1 ± +2 2± + 1 2± + 2
0 1 2 2±
±

0 0 0 0 0 0 0 0 0 0
±+1 ±+2 2± + 1 2± + 2
1 0 1 2 2±
±
2± + 2 2± + 1 ±+2 ±+1
2 0 2 1 2± ±
±+2 2± + 2 ±+1 2± + 1
0 2± 2 1
± ±
±+1 ±+1 2± + 2 ±+2 2± + 1
0 2± 1 2 ±
±+2 ±+2 2± + 1 2± + 2 ±+1
0 1 2± 2
±
2± + 1 ±+1 2± + 2 ±+2
2± 0 2± 1 2
±
2± + 1 2± + 1 ± +2 ±+1 2± + 2
0 2 2± 1
±
2± + 2 2± + 2 ± +1 2± + 1 ±+2
0 2 1 2±
±


11.5. x 3 + x + 4. 11.7. Impossible.
11.9. x 2 + 2. 11.11. x 4 ’ 16x 2 + 16.
11.13. 8x 6 ’ 9. 11.17. 3.
318 ANSWERS TO THE ODD-NUMBERED EXERCISES


11.19. 2. 11.21. 2.
∞. √ 11.25. ∞.
11.23. √
(1 ’ 2 + 4)/3. 11.29. ’(1 + 6ω)/31.
3 3
11.27.
± + ±.
4
11.31. 11.33. 2; not a ¬eld.
11.35. 7; a ¬eld 11.37. 0; a ¬eld.
11.39. 0; not a ¬eld.
m = 2, 4, p r or 2p r , where p is an odd prime; see Weiss [12, Th. 4“6“10].
11.43.
√ √
± = 2 + ’3.
11.47.
11.49. All elements of GF (32) except 0 and 1 are primitive.
x 3 + x + 1.
11.51. 11.57. No solutions.
x = 1 or ± + 1.
11.59. 11.63. 5.
11.65. The output has cycle length 7 and repeats the sequence 1101001, starting
at the right.


CHAPTER 12

12.1.
a b c d e f g
b c d e f g a
c d e f g a b
d e f g a b c
e f g a b c d
f g a b c d e
g a b c d e f

12.3. Use GF(8) = {0, 1, ±, 1 + ±, ± 2 , 1 + ± 2 , ± + ± 2 , 1 + ± + ± 2 } where
± 3 = ± + 1.
12.7. No.
12.9.
Week
1 2 3 4
“ “ “ “
’ A B C D

Shelf B C D A

height C D A B
’ D A B C

A, B, C, and D are the four brands of cereal.
ANSWERS TO THE ODD-NUMBERED EXERCISES 319


12.11.
Week
1 2 3 4 5
“ “ “ “ “

M 0
A B C D

T 0
B C D A

W 0
C D A B

T 0
D A B C

F 0 A B C D

A, B, C, and D are the four different types of music, and 0 refers to
no music.
12.15. y = ±x. 12.17. y = (± + 2)x + 2±.
12.21. 1.
12.23.
1 6 11 16 1 6 11 16
12 15 2 5 15 12 5 2
14 9 8 3 8 3 14 9
7 4 13 10 10 13 4 7


12.25.
1 10 19 28 37 46 55 64
35 44 49 58 7 16 21 30
29 22 15 8 57 50 43 36
63 56 45 38 27 20 9 2
52 59 34 41 24 31 6 13
18 25 4 11 54 61 40 47
48 39 62 53 12 3 26 17
14 5 32 23 42 33 60 51


CHAPTER 13

13.1. Constructible. 13.3. Constructible.
13.5. Not constructible. 13.7. No.
13.11. Yes. 13.13. No.

π π
= ’
13.15. Yes; . 13.23. Yes.
21 43 7
13.25. No. 13.27. No.
13.29. Yes. 13.31. No.
13.33. Yes.


CHAPTER 14
14.1. 010, 001, 111.
320 ANSWERS TO THE ODD-NUMBERED EXERCISES


14.3. The checking is done modulo 9, using the fact that any integer is congruent
to the sum of its digits modulo 9.
14.5. (1 2 3 4 5 6 7 8 9 10) modulo 11. It will detect one error but not
correct any.
14.7. 000000, 110001, 111010, 001011, 101100, 011101, 010110, 100111.
14.9. 101, 001, 100.
14.11. Minimum distance = 3. It detects two errors and corrects one error.
« 
1 0 0 0 0 0 0 1 0
¬0 1·
1 0 0 0 0 1 0
¬ ·
H = ¬0 1·.
0 1 0 0 1 0 1
0 0
0 0 1 0 0 1 0
0 0 0 0 1 1 0 0 1
«
1001
14.13. GT = 1 1 1 1 , H =  0 1 0 1  .
0011
« 
101001000
¬0 1 0 1 0 0 1 0 0·
14.15. GT = ¬ ·
0 0 1 0 1 0 0 1 0,
101100001
« 
100001001
¬0 1 0 0 0 0 1 0 0·
¬ ·
H = ¬0 0 1 0 0 1 0 1 1·.
0 0 0 1 0 0 1 0 1
000010010
14.17. 110, 010, 101, 001, 011.
14.19.
Syndrome Coset Leader
00 000
01 010
10 100
11 001
ANSWERS TO THE ODD-NUMBERED EXERCISES 321


14.21.
Coset Coset Coset
Syndrome Leader Syndrome Leader Syndrome Leader
00000 000000000 01011 000010100 10110 000000001
00001 000010000 01100 011000000 10111 000010001
00010 000100000 01101 010000010 11000 110000000
00011 000110000 01110 001000100 11001 000000111
00100 001000000 01111 000000110 11010 100000100
00101 000000010 10000 100000000 11011 000001110
00110 001100000 10001 000001010 11100 000000101
00111 000100010 10010 100100000 11101 000010101
01000 010000000 10011 000000011 11110 000001100
01001 010010000 10100 000001000 11111 000011100
01010 000000100 10101 000011000
14.23. (a) 56; (b) 7; (c) 27 = 128, (d) 8/9; (e) it will detect single, double, triple,
and any odd number of errors; (f) 1.
14.25. No.
14.29. x 8 + x 4 + x 2 + x + 1 and x 10 + x 9 + x 8 + x 6 + x 5 + x 2 + 1.
14.31. x 5 + x 4 + x 3 + x 2 + 1.
INDEX


Abel, N. H., 1, 47, 48 Biconditional, 18
Abelian group, 48 Bijective, 50, 63
¬nite, 92 Binary code, 266
Absorption laws, 15 Binary operation, 2
Abstract algebra, 2 Binary symmetric channel, 266
Action of a group, 96 Binomial theorem, 178
Adder, full, 35 coef¬cients, 129
half, 34 Boole, G., 7
modulo 2, 28 Boolean algebra, 7, 14, 25
serial, 152 isomorphism, 39
Addition modulo m, 84 morphism, 39
Adjoining an element, 220 representation theorem, 39
Af¬ne plane, 242 Boolean expression, 26
Algebra, abstract, 2 Boolean ring, 158, 176
boolean, 7, 14, 25 Bose, R. C., 242, 284
classical, 1 Bridge circuit, 43
modern, 2 Burnside, W., 125
Algebraic numbers, 221, 233 Burnside™s theorem (lemma), 125
Algebraic structure, 4
Algebra of sets, 7 Cancellation, 53
Alternating group, 70, 81, 85, 88 Cantor™s theorem, 41
AND gate, 36 Carrol, L., 41
Angle trisection, 251, 257 Casting out nines, 288
Antisymmetry, 23 Cauchy sequences, 6
Archimedean solids, 121 Cayley, A., 47, 71
Archimedes, 258 Cayley™s theorem, 71
Associativity, 3, 14, 48, 137, 155 CD, 264
Atom, 26 Center of a group, 74
Automorphism, 75, 151 Characteristic of a ring, 226
Frobenius, 235 Check digit, 266
Axiomatic method, 295 Chinese remainder theorem, 198
Axioms, 295 Circle group, 88, 98
Axis of symmetry, 51 Circle squaring, 251, 259
Circuit, see Switching circuits
BCH code, 284 Classical algebra, 1
Benzene, 126 Closed switch, 19

Modern Algebra with Applications, Second Edition, by William J. Gilbert and W. Keith Nicholson
ISBN 0-471-41451-4 Copyright ™ 2004 John Wiley & Sons, Inc.
323
324 INDEX

Closure of an operation, 3, 48 Cycle, 64
disjoint, 66
Code, 264
Cyclic code, 291
BCH, 284
Cyclic group, 56
binary, 266
Cyclic monoid, 139, 150
cyclic, 292
Cyclic subgroup, 57
error-correcting, 265
Cyclotomic polynomial, 195
error-detecting, 264
Cylinder, n-agonal, 116
linear, 276
(n, k)-, 266
parity check, 267, 274 Da Vinci, L., 109
polynomial, 270 Decode, 280
repeating, 268 Dedekind cuts, 6
Code rate, 267 Degree, of an extension, 219
Code word, 266 of a polynomial, 166
Coef¬cient, 166 Delta function, 172
Colorings, 128 De Morgan™s laws, 15
Common divisor, 184, 299 Derivative, 292
multiple, 184, 303 Detecting errors, 280
Commutativity, 3, 14, 48, 148, 167 Determinant, 110, 112, 287
Commutator, 101 Diagram, state, 143
Complement, 8, 14 tree, 148
Complex numbers, 4, 223 Venn, 8, 32
Complex roots, 191 Difference, 9
Composition of relations, 150 symmetric, 9
of functions, 49 Dihedral group, 58, 90
of relations, 150 Dirac delta function, 172
Conclusion, 293 Direct product, 2, 91, 164
Concatenation, 140 Direct sum, 91
Disjoint cycles, 66
Conditional, 17
Disjunctive normal form, 30
Cone, n-agonal, 116
Distributions, 175
Congruence, 77, 79, 145
Distributivity, 4, 14, 156
linear, 197
Division, 184, 299
Congruence class, 77, 145
Division algorithm, 180, 181, 298
Conjugate, 191
Divisor of zero, 159
Conjunctive normal form, 45
Dodecahedron, 112
Constant polynomial, 167
rotation group, 114
Constructible number, 252
Domain, 172
Constructible point, 252
left Ore, 172
Construction of polygons, 259
Domain, integral, 159
Contradiction, 17
Duality in boolean algebras, 15
proof by, 294
Duality of regular solids, 112
Converse, 295
Duplication of the cube, 251, 256
Convolution fraction, 175
D¨ rer, A., 246
u
Convolution of functions, 173
DVD, 264
of sequences, 168
Coset, 79, 82
Coset leader, 280 Eigenvalue, 108, 109, 111
Countable, 221 Eigenvector, 108, 110
Counterexample, 295 Eisenstein™s criterion, 194
Cross-ratio, 101 Element of a set, 7
Crystalline lattice, 120 Empty set, 7
Crystallographic group, 120 Encode, 266, 280
Cube, duplication of, 251, 256 Encoding matrix, 276
Cube, rotation group of, 114 Endomorphism ring, 178
INDEX 325

Equivalence class, 77 Free, group, 75
monoid, 140
Equivalence, logical, 295
semigroup, 140
Equivalence relation, 5, 77
Full adder, 35
Equivalent circuits, 20
Frobenius automorphism, 235
Error-correcting code, 264
Function, bijective, 50, 63
Error-detecting code, 264
composition of, 49
Error polynomial, 275
delta, 172
Euclidean algorithm, 185, 300
generalized, 175
Euclidean group, 104
Heaviside, 175
Euclidean ring, 181
impulse, 172
Euclid™s lemma, 302
injective, 49
Euclid™s theorem, 298
inverse, 49
Euler, L., 103, 236, 242, 260
one-to-one, 49
Euler φ-function, 103
onto, 50
Even permutation, 68
surjective, 49
Exclusive OR, 10, 29
transition, 142
Extension ¬eld, 218
Fundamental theorem of, algebra, 6, 190
degree of, 219
arithmetic, 187
¬nite, 219
´
Galois, E., 6, 47, 225
Factor, 184, see also Quotient Galois ¬eld, 6, 227
Factor theorem, 182 Galois theory, 47,
Faithful action of a group, 96 Gate, 36
Faithful representation, 109 Gauss, C. F., 6, 190, 260
Feedback shift register, 231, 272 Gaussian integers, 72, 183
Fermat, P., 103, 260 Gauss™ lemma, 193
Fermat primes, 260 Generalized function, 175
Fermat™s last theorem, 305 General linear group, 107
Field, 4, 160 Generator, group, 56
¬nite, 6, 45, 225 idempotent, 292
Galois, 6, 227 matrix, 276
primitive element in, 229 of a monoid, 139
skew, 172 polynomial, 270
Field extension, 218 Geometry, 242
Field of, convolution fractions, 175 Goldbach conjecture, 304
fractions, 170 Graeco-Latin square, see Orthogonal latin
quotients, 170 squares
rational functions, 172, 221 Greatest common divisor, 21, 184, 299
Fifteen puzzle, 74 Greatest lower bound, 25
Finite, abelian group, 92 Group, 48
extension, 219 abelian, 48
¬eld, 6, 45, 225 alternating, 70, 82, 86, 88
geometry, 242 automorphism, 75
group, 56 center, 74
groups in three dimensions, 116 circle, 88, 98
groups in two dimensions, 109 commutative, 48
Finite-state machine, 142 crystallographic, 120
First isomorphism theorem, 87, 210 cyclic, 56
Fixed points, 125 dihedral, 58, 90
Flip-¬‚op, 46 euclidean, 104
Formal power series, 169 factor, 83
Fractional differentiation and integration, 175 ¬nite, 56
Fractions, ¬eld of, 170 free, 75
left, 172 general linear, 107
326 INDEX

Group (continued) Identity, 4, 48, 137
function, 49
generator, 56
If and only if, 18, 295
icosahedral, 115
Image, 87
in¬nite, 56
Implication, 17, 293
isomorphism, 60
Improper rotation, 52, 58, 108
Klein, 52, 90
Impulse functions, 172
matrix, 107
Inclusion, 7
metabelian, 102
Index of a subgroup, 80
morphism, 60
Induction, 296
noncommutative, 59
extended, 297
octahedral, 114
strong, 297
of a polynomial, 80
In¬nite group, 56
of a rectangle, 53, 55
Information rate, 267
of a square, 89
Injective, 49, 86
of low order, 94
Input values, 142
of prime order, 80
Integers, 5, 156, 296
order, 56
gaussian, 72, 183
orthogonal, 105
Integers modulo m, 78
permutation, 50
Integral domain, 159
quaternion, 73, 95, 107
Interlacing shuf¬‚e, 74
quotient, 83
International standard book number, 289
simple, 86
Intersection of sets, 8
special orthogonal, 98, 108
Inverse, 3, 48, 188
special unitary, 108
function, 49
sporadic, 86
Inversion theorem, 50
symmetric, 50, 63
Invertible element, 3, 48, 188
symmetries, 51
Irrational roots, 192
tetrahedral, 113
Irreducible, 189
translation, 49, 104
polynomial, 190
trivial, 49
Isometry, 51, 112
unitary, 108
Isomorphism, 5
Group acting on a set, 96
boolean algebra, 39
Group isomorphism, 60
group, 60
Group morphism, 60
monoid, 141
ring, 162
Half adder, 34 Isomorphism theorems for groups, 87, 101,
Hamming distance, 268 102
Heaviside, O., 172, 175 Isomorphism theorems for rings, 210, 215
Homomorphism, see Morphism
Hypothesis, 293 Join, 14
Juxtaposition, 140
Icosahedral group, 115
Kernel, 86
Icosahedron, 112
Klein, F., 52
rotation group, 114
Klein 4-group, 52, 90
Ideal, 204
Kronecker, L., 5
left, 217
maximal, 216
prime, 216 Lagrange, J., 79
principal, 204 Lagrange™s theorem, 80
Ideal crystalline lattice, 120 Latin square, 236
Idempotent element, 179 orthogonal, 239
generator, 292 Lattice, 25
laws, 15 crystalline, 120
INDEX 327

Least common multiple, 21, 184, 303 Monoid morphism, 141
Monoid of a machine, 145
Least upper bound, 25
Monoid of transformations, 138
Left coset, 82
Morphism, 5
Left ideal, 217
boolean algebra, 39
Linear code, 276
group, 60
cyclic, 291
monoid, 141
Linear congruences, 197,
ring, 172
Linear transformation, 5, 104
Morphism theorem for, groups, 87, 101, 102
Lines in a geometry, 242
monoids, 150
Length of a vector, 105
rings, 210
Local ring, 216
Mutually orthogonal squares, 239
Logically equivalent, 17, 295
Logic of propositions, 16
NAND gate, 28, 36
Necklace problems, 126
Machine, 142
Network, see Switching circuits
monoid of, 145
Nilpotent element, 216
parity checker, 143
Nonsingular matrix, 4
semigroup of, 142
NOR gate, 28, 36
Magic square, 247
Normal form, conjunctive, 45
Mathematical theory, 295
disjunctive, 30
Matrix
Normal subgroup, 82
eigenvalue of, 108
NOT gate, 36
eigenvector of, 108
encoding, 276
Octahedral group, 114
generator, 276
Octahedron, 112
nonsingular, 4
rotation group, 114
orthogonal, 105
Odd permutation, 68
parity check, 278
One-to-one, 49
unitary, 108
One-to-one correspondence, 50
Matrix group, 107
Onto, 50
Matrix representation, 109
Open switch, 19
of codes, 276
Operation, 2
Matrix ring, 166
Operational calculus, 172
Maximal ideal, 216
Operator, 175
Meet, 14
Orbit, 65, 78, 97
Metabelian group, 102
Order of a group, 56
Metric, 291
Order of an element, 56
Mikusinski, J., 173
OR gate, 36
Modern algebra, 2
Orthogonal group, 105
Modular representation, 200
Orthogonal latin squares, 239
Modulo m, 78
Orthogonal matrix, 105
Modulo 2 adder, 28
Output, 142
Monic polynomial, 234
Monoid, 137
automorphism, 151 Parallel circuit, 19
cyclic, 139, 150 Parallelism, 242
free, 140 Parity check, 278
generator, 139 code, 267, 274
morphism, 141 machine, 143
morphism theorem, 150 matrix, 278
quotient, 145 Parity of a permutation, 69
representation theorem, 150 Partial order, 24
transformations, 138 Partition, 77
Monoid isomorphism, 141 Pattern recognition, 147
328 INDEX

Permutation, 50, 63 Quaternion, group, 73, 95, 107
ring, 172, 177
even, 68
Quotient, 181, 299
odd, 68
¬eld of, 170
parity, 69
group, 83
Permutation group, 50, 63
monoid, 145
Phi function, 103
ring, 206
Plane, af¬ne, 242
set, 72
projective, 245
structure, 5
Points of a geometry, 242
Pole of a rotation, 116
P´ lya, G., 125, 134
o Radical of a ring, 216
P´ lya-Burnside enumeration, 124
o Rational functions, 172, 221
Polygons, construction, 259 Rational number, 6, 48, 78, 170
Polyhedron, dual, 112 Rational roots theorem, 192
Polynomial, 166 Real numbers, 6, 48, 156
coef¬cients, 166 Real numbers modulo one, 88
constant, 167 Real projective space, 102
cyclotomic, 195 Rectangle, symmetries of, 53, 55
degree, 166 Reducible, 190
equality, 166 Redundant digits, 267
group of symmetries, 74 Re¬‚exivity, 23, 77
irreducible, 189, 190 Register, shift, 231, 272
monic, 234 Regular n-gon, rotations, 58
primitive, 231, 275 Regular polygons, construction, 259
reducible, 190 Regular solid, 112
zero, 166 Relation, 76
Polynomial code, 270 composition of, 150
Polynomial equations, 1, 47 congruence, 77, 145
Polynomial representation of codes, 270 equivalence, 77
Polynomial ring, 167 partial order, 24
Poset, 24 Relatively prime, 301
Positive integers, 3 Remainder, 181, 299
Power series, 169 Remainder theorem, 182
Power set, 8 Repeating code, 268
Prime, 189, 294, 297 Representation, faithful, 109
factorization theorem, 21, 302 matrix, 109
Fermat, 260 modular, 200
ideal, 216 residue, 200
Prime order group, 80 Representation theorem for, boolean
Primitive element, 229 algebras, 39, 40
Primitive polynomial, 231, 275 groups, 71
Principal ideal, 204 monoids, 150
Principal ideal ring, 205 Representative of a coset, 79
Product group, 91 Residue representation, 200
Product ring, 164 Right coset, 79
Projective plane, 245 Ring, 155
Projective space, 102 boolean, 158, 176
Proof, 293 characteristic, 226
by contradiction, 294 commutative, 156
by reduction to cases, 293 endomorphism, 178
direct, 293 euclidean, 181
Proper rotation, 52, 58, 108 factor, 206
symmetry, 52 ¬eld, 160
Propositional calculus, 18 ideal of, 204
INDEX 329

integral domain, 159 Solids, regular, 112
local, 216 Space, projective, 102
matrix, 166 Special orthogonal group, 98, 108
morphism, 172 Special unitary group, 108
morphism theorem, 210 Sphere, 116
nontrivial, 159 Sporadic groups, 86
polynomial, 166 Square, latin, 236
principal ideal, 205 magic, 247
product, 164 orthogonal latin, 239
quotient, 206 Square-free integer, 22, 302
radical of, 216 Squaring the circle, 251, 259
simple, 217 Stabilizer, 97
subring of, 161 Standard basis, 104
trivial, 159 Standard matrix, 105, 164
Ring isomorphism, 162 State, 142
Ring morphism, 162, 210
diagram, 143
Ring of, formal power series, 169
Step function, 175
matrices, 166
Stone™s representation theorem, 40
polynomials, 166
Straight-edge and compass constructions, 251
quaternions, 172, 177
Structure, algebraic, 4
sequences, 168
Structure, quotient, 5
Roots, 183
Sub¬eld, 218
complex, 190, 191
smallest, 220
irrational, 191
Subgroup, 54
rational, 192
commutator, 101
Rotations, 52, 58, 108
cyclic, 57
Rotations of, a cube, 114
index of, 80
a dodecahedron, 114
normal, 92
an icosahedron, 114
Submonoid, 150
an n-gon, 58
Subring, 161
an octahedron, 114
Subset, 7
a tetrahedron, 113
Substructure, 4
Ruler and compass, 251
Sum, direct, 91
Surjective, 49
Second isomorphism theorem, 102, 215
Switch, 19
Semigroup, 137
Switching circuits, 20
free, 140
bridge, 43
Semigroup of a machine, 145
number of, 130
Sequences, ring of, 168
n-variable, 28
Serial adder, 152
series-parallel, 28
Series, power, 169
Switching function, 27
Series circuit, 19
Sylow theorems, 85
Series-parallel circuit, 20
Symmetric condition, 23, 77
Set, 7
Symmetric difference, 9, 28
Sets, algebra of, 7
Symmetric group, 50, 63
Shannon, C. E., 267
Symmetries of a
Shift register, 231, 272
¬gure, 3, 51
Shuf¬‚e, interlacing, 74
polynomial, 74
Simple, group, 86
rectangle, 53, 55
ring, 217
set, 50
Simpli¬cation of circuits, 26
square, 89
Skew ¬eld, 172
Symmetry, proper, 52
Smallest sub¬eld, 220
Solids, Archimedean, 121 Syndrome, 280
330 INDEX

Table, 3 Unary operation, 2, 8, 14
truth, 16 Underlying set, 4
Tarry, G., 242 Union of sets, 8
Tautology, 17 Unique factorization theorem, 189
Tetrahedral group, 113 Unitary group, 108
Tetrahedron, 112 Unit element, 188. See also Identity;
rotation group, 113 Invertible element
Third isomorphism theorem, 102, 215 Unity, see Identity
Titchmarsh™s theorem, 174 Universal bounds, 25
Transcendental, 221
Transformation monoid, 138
Vandermonde determinant, 287
Transient conditions in circuits, 49
Vector space, 5
Transistor gates, 36
Venn diagram, 8, 32
Transition function, 142
Transitivity, 23, 77
Translation, 49, 104 Well ordering axiom, 296
Transpose of a matrix, 104 Wilson™s theorem, 202
Transposition, 68 Words, 140
Tree diagram, 148
Trisection of an angle, 251, 257
Zero, 14, 155
Trivial group, 49
Zero divisor, 159
Trivial ring, 159
Zero polynomial, 166
Truth table, 16
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