Abstract Algebra Theory and Applications - Computer Science ...

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ContentsPrefacevii0 Preliminaries 10.1 A Short Note on Proofs . . . . . . . . . . . . . . . . . . . . . 10.2 Sets and Equivalence Relations . . . . . . . . . . . . . . . . . 41 The Integers 221.1 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . 221.2 The Division Algorithm . . . . . . . . . . . . . . . . . . . . . 262 Groups 352.1 The Integers mod n and Symmetries . . . . . . . . . . . . . . 352.2 Definitions and Examples . . . . . . . . . . . . . . . . . . . . 402.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 Cyclic Groups 563.1 Cyclic Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 The Group C ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3 The Method of Repeated Squares . . . . . . . . . . . . . . . . 644 Permutation Groups 724.1 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . 734.2 The Dihedral Groups . . . . . . . . . . . . . . . . . . . . . . . 815 Cosets and Lagrange’s Theorem 895.1 Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Lagrange’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 925.3 Fermat’s and Euler’s Theorems . . . . . . . . . . . . . . . . . 94x
CONTENTSxi6 Introduction to Cryptography 976.1 Private Key Cryptography . . . . . . . . . . . . . . . . . . . . 986.2 Public Key Cryptography . . . . . . . . . . . . . . . . . . . . 1017 Algebraic Coding Theory 1087.1 Error-Detecting and Correcting Codes . . . . . . . . . . . . . 1087.2 Linear Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.3 Parity-Check and Generator Matrices . . . . . . . . . . . . . 1217.4 Efficient Decoding . . . . . . . . . . . . . . . . . . . . . . . . 1288 Isomorphisms 1388.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . 1388.2 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . 1439 Homomorphisms and Factor Groups 1529.1 Factor Groups and Normal Subgroups . . . . . . . . . . . . . 1529.2 Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . 1559.3 The Isomorphism Theorems . . . . . . . . . . . . . . . . . . . 16210 Matrix Groups and Symmetry 17010.1 Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 17010.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17911 The Structure of Groups 19011.1 Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . 19011.2 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . 19512 Group Actions 20312.1 Groups Acting on Sets . . . . . . . . . . . . . . . . . . . . . . 20312.2 The Class Equation . . . . . . . . . . . . . . . . . . . . . . . 20712.3 Burnside’s Counting Theorem . . . . . . . . . . . . . . . . . . 20913 The Sylow Theorems 22013.1 The Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . 22013.2 Examples and Applications . . . . . . . . . . . . . . . . . . . 22414 Rings 23214.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23214.2 Integral Domains and Fields . . . . . . . . . . . . . . . . . . . 23714.3 Ring Homomorphisms and Ideals . . . . . . . . . . . . . . . . 23914.4 Maximal and Prime Ideals . . . . . . . . . . . . . . . . . . . . 243
Page 1 and 2: Abstract AlgebraTheory and Applicat
Page 3 and 4: PrefaceThis text is intended for a
Page 5: PREFACEixappears at the end of each
Page 9 and 10: 0PreliminariesA certain amount of m
Page 11 and 12: 0.1 A SHORT NOTE ON PROOFS 3ax 2 +
Page 13 and 14: 0.2 SETS AND EQUIVALENCE RELATIONS
Page 15: 0.2 SETS AND EQUIVALENCE RELATIONS
Page 18 and 19: 10 CHAPTER 0 PRELIMINARIESnot all f
Page 20 and 21: 12 CHAPTER 0 PRELIMINARIESThis is a
Page 22 and 23: 14 CHAPTER 0 PRELIMINARIESgiven by
Page 24 and 25: 16 CHAPTER 0 PRELIMINARIESandB =the
Page 26 and 27: 18 CHAPTER 0 PRELIMINARIESIf we con
Page 28 and 29: 20 CHAPTER 0 PRELIMINARIES(e) If g
Page 30 and 31: 1The IntegersThe integers are the b
Page 32 and 33: 24 CHAPTER 1 THE INTEGERSwhere a an
Page 34 and 35: 26 CHAPTER 1 THE INTEGERSEvery good
Page 36 and 37: 28 CHAPTER 1 THE INTEGERSThe Euclid
Page 38 and 39: 30 CHAPTER 1 THE INTEGERSTheorem 1.
Page 40 and 41: 32 CHAPTER 1 THE INTEGERS9. Use ind
Page 42 and 43: 34 CHAPTER 1 THE INTEGERSProgrammin
Page 44 and 45: 36 CHAPTER 2 GROUPSZ n . Consider t
Page 46 and 47: 38 CHAPTER 2 GROUPSSymmetriesA symm
Page 48 and 49: 40 CHAPTER 2 GROUPSTable 2.2. Symme
Page 50 and 51: 42 CHAPTER 2 GROUPSand 4. We define
Page 52 and 53: 44 CHAPTER 2 GROUPSis the inverse o
Page 54 and 55: 46 CHAPTER 2 GROUPS1. mg + ng = (m
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48 CHAPTER 2 GROUPSnecessarily obta
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50 CHAPTER 2 GROUPS3. Write out Cay
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52 CHAPTER 2 GROUPS32. Find all the
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54 CHAPTER 2 GROUPS0 50000 30042 6F
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3Cyclic GroupsThe groups Z and Z n
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58 CHAPTER 3 CYCLIC GROUPS✦ ❛
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60 CHAPTER 3 CYCLIC GROUPS3.2 The G
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62 CHAPTER 3 CYCLIC GROUPSanda = r
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64 CHAPTER 3 CYCLIC GROUPSTheorem 3
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66 CHAPTER 3 CYCLIC GROUPSk multipl
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68 CHAPTER 3 CYCLIC GROUPS4. Find t
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70 CHAPTER 3 CYCLIC GROUPS27. If g
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4Permutation GroupsPermutation grou
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74 CHAPTER 4 PERMUTATION GROUPSthis
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76 CHAPTER 4 PERMUTATION GROUPSExam
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78 CHAPTER 4 PERMUTATION GROUPSExam
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80 CHAPTER 4 PERMUTATION GROUPSProo
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82 CHAPTER 4 PERMUTATION GROUPSresu
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84 CHAPTER 4 PERMUTATION GROUPSand
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86 CHAPTER 4 PERMUTATION GROUPS(a)
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88 CHAPTER 4 PERMUTATION GROUPS(b)
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90 CHAPTER 5 COSETS AND LAGRANGE’
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98 CHAPTER 6 INTRODUCTION TO CRYPTO
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100 CHAPTER 6 INTRODUCTION TO CRYPT
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7Algebraic Coding TheoryCoding theo
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110 CHAPTER 7 ALGEBRAIC CODING THEO
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8IsomorphismsMany groups may appear
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140 CHAPTER 8 ISOMORPHISMSab ≠ ba
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142 CHAPTER 8 ISOMORPHISMSThe addit
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144 CHAPTER 8 ISOMORPHISMSthat is,
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146 CHAPTER 8 ISOMORPHISMSTherefore
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148 CHAPTER 8 ISOMORPHISMSWe will l
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150 CHAPTER 8 ISOMORPHISMS20. Prove
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9Homomorphisms and FactorGroupsIf H
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154 CHAPTER 9 HOMOMORPHISMS AND FAC
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10Matrix Groups andSymmetryWhen Fel
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172 CHAPTER 10 MATRIX GROUPS AND SY
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11The Structure of GroupsThe ultima
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192 CHAPTER 11 THE STRUCTURE OF GRO
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204 CHAPTER 12 GROUP ACTIONSThe ele
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206 CHAPTER 12 GROUP ACTIONSand the
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208 CHAPTER 12 GROUP ACTIONSExample
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210 CHAPTER 12 GROUP ACTIONSLemma 1
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212 CHAPTER 12 GROUP ACTIONSinduces
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214 CHAPTER 12 GROUP ACTIONSSwitchi
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216 CHAPTER 12 GROUP ACTIONSTable 1
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218 CHAPTER 12 GROUP ACTIONS10. Fin
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13The Sylow TheoremsWe already know
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222 CHAPTER 13 THE SYLOW THEOREMSHe
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224 CHAPTER 13 THE SYLOW THEOREMSpo
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226 CHAPTER 13 THE SYLOW THEOREMSbe
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228 CHAPTER 13 THE SYLOW THEOREMSSu
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230 CHAPTER 13 THE SYLOW THEOREMS(e
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14RingsUp to this point we have stu
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234 CHAPTER 14 RINGSExample 3. We c
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236 CHAPTER 14 RINGS3. (−a)(−b)
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238 CHAPTER 14 RINGSProposition 14.
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240 CHAPTER 14 RINGSa ring homomorp
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242 CHAPTER 14 RINGSTheorem 14.9 Le
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244 CHAPTER 14 RINGSTheorem 14.15 L
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246 CHAPTER 14 RINGSthe Senate is n
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248 CHAPTER 14 RINGShas a solution.
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250 CHAPTER 14 RINGSMultiplying the
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252 CHAPTER 14 RINGS11. Prove that
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254 CHAPTER 14 RINGS34. Let R be a
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15PolynomialsMost people are fairly
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258 CHAPTER 15 POLYNOMIALSExample 1
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260 CHAPTER 15 POLYNOMIALSProof. Su
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262 CHAPTER 15 POLYNOMIALSm ≥ n.
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264 CHAPTER 15 POLYNOMIALSPropositi
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268 CHAPTER 15 POLYNOMIALSwhere eac
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272 CHAPTER 15 POLYNOMIALS(c) p(x)
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274 CHAPTER 15 POLYNOMIALSAdditiona
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276 CHAPTER 15 POLYNOMIALS(a) x 4
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278 CHAPTER 16 INTEGRAL DOMAINSelem
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280 CHAPTER 16 INTEGRAL DOMAINSIt i
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282 CHAPTER 16 INTEGRAL DOMAINSLet
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284 CHAPTER 16 INTEGRAL DOMAINSCons
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286 CHAPTER 16 INTEGRAL DOMAINSEucl
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288 CHAPTER 16 INTEGRAL DOMAINSin D
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290 CHAPTER 16 INTEGRAL DOMAINSCoro
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292 CHAPTER 16 INTEGRAL DOMAINS7. L
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17Lattices and BooleanAlgebrasThe a
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296 CHAPTER 17 LATTICES AND BOOLEAN
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18Vector SpacesIn a physical system
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314 CHAPTER 18 VECTOR SPACES5. −(
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316 CHAPTER 18 VECTOR SPACESProposi
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318 CHAPTER 18 VECTOR SPACES3. If S
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320 CHAPTER 18 VECTOR SPACES(d) Let
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19FieldsIt is natural to ask whethe
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324 CHAPTER 19 FIELDS· 0 1 α 1 +
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326 CHAPTER 19 FIELDSExample 5. We
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328 CHAPTER 19 FIELDSfor b i and c
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330 CHAPTER 19 FIELDSLetu = ∑ i,j
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332 CHAPTER 19 FIELDSwhere each fie
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334 CHAPTER 19 FIELDSLet F be a fie
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336 CHAPTER 19 FIELDSTheorem 19.19
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338 CHAPTER 19 FIELDSlength αβ. A
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340 CHAPTER 19 FIELDSbe the equatio
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342 CHAPTER 19 FIELDScosine,4α 3
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344 CHAPTER 19 FIELDS11. Let p(x) b
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20Finite FieldsFinite fields appear
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348 CHAPTER 20 FINITE FIELDSextensi
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350 CHAPTER 20 FINITE FIELDSGF(p 24
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352 CHAPTER 20 FINITE FIELDSMessage
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354 CHAPTER 20 FINITE FIELDSThe add
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356 CHAPTER 20 FINITE FIELDSExample
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358 CHAPTER 20 FINITE FIELDSand g(x
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360 CHAPTER 20 FINITE FIELDS(3) ⇒
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362 CHAPTER 20 FINITE FIELDS22. Sho
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21Galois TheoryA classic problem of
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366 CHAPTER 21 GALOIS THEORYthe Gal
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368 CHAPTER 21 GALOIS THEORYof G(E/
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370 CHAPTER 21 GALOIS THEORYin K. A
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372 CHAPTER 21 GALOIS THEORYThe pro
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374 CHAPTER 21 GALOIS THEORY{id, σ
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376 CHAPTER 21 GALOIS THEORYThe ker
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378 CHAPTER 21 GALOIS THEORYfor res
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380 CHAPTER 21 GALOIS THEORYis a no
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382 CHAPTER 21 GALOIS THEORYand onc
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384 CHAPTER 21 GALOIS THEORY(a) x 5
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386 CHAPTER 21 GALOIS THEORY[3] Fra
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388 NOTATIONSymbol Description Page
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390 NOTATIONSymbol Description Page
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392 HINTS AND SOLUTIONSConversely,
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394 HINTS AND SOLUTIONS4. (a)(c)( 1
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396 HINTS AND SOLUTIONS7. (a) (0000
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398 HINTS AND SOLUTIONS9. For any h
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400 HINTS AND SOLUTIONS2. The four
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402 HINTS AND SOLUTIONSChapter 17.
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404 HINTS AND SOLUTIONSChapter 19.
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GNU Free Documentation LicenseVersi
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408 GFDL LICENSEappear in the title
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410 GFDL LICENSEH. Include an unalt
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412 GFDL LICENSEply to the other wo
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IndexG-equivalent, 205G-set, 203nth
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416 INDEXEquivalence relation, 15Eu
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418 INDEXKlein, Felix, 46, 170, 245
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420 INDEXnormalizer of, 222proper,
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