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386 CHAPTER 21 GALOIS THEORY[3] Fraleigh, J. B. A First Course in Abstract Algebra, 4th ed. Addison-Wesley,Reading, MA, 1989.[4] Gaal, L. Classical Galois Theory with Examples. 2nd ed. Chelsea, New York,1973.[5] Garling, D. J. H. A Course in Galois Theory. Cambridge University Press,Cambridge, 1986.[6] Kaplansky, I. Fields and Rings. 2nd ed. University of Chicago Press, Chicago,1972.[7] Rothman, T. “The Short Life of Évariste Galois,” Scientific American, April1982, 136–49.
NotationThe following table defines the notation used in this book. Page numbersrefer to the first appearance of each symbol.Symbol Description Pagea ∈ A a is in the set A 4N the natural numbers 5Z the integers 22Q the rational numbers 5R the real numbers 5C the complex numbers 5A ⊂ B A is a subset of B 5∅ the empty set 5A ∪ B union of sets A and B 5A ∩ B intersection of sets A and B 5A ′ complement of the set A 6A \ B difference between sets A and B 6A × B Cartesian product of sets A and B 8A n A × · · · × A (n times) 8id identity mapping 12f −1 inverse of the function f 13a ≡ b (mod n) a is congruent to b modulo n 17n!( )n factorial 24nbinomial coefficient n!/(k!(n − k)!) 24km | n m divides n 27gcd(m, n) greatest common divisor of m and n 27387
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Abstract AlgebraTheory and Applicat
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PrefaceThis text is intended for a
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PREFACEixappears at the end of each
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CONTENTSxi6 Introduction to Cryptog
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0PreliminariesA certain amount of m
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0.1 A SHORT NOTE ON PROOFS 3ax 2 +
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0.2 SETS AND EQUIVALENCE RELATIONS
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0.2 SETS AND EQUIVALENCE RELATIONS
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10 CHAPTER 0 PRELIMINARIESnot all f
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12 CHAPTER 0 PRELIMINARIESThis is a
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14 CHAPTER 0 PRELIMINARIESgiven by
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16 CHAPTER 0 PRELIMINARIESandB =the
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18 CHAPTER 0 PRELIMINARIESIf we con
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20 CHAPTER 0 PRELIMINARIES(e) If g
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1The IntegersThe integers are the b
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24 CHAPTER 1 THE INTEGERSwhere a an
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26 CHAPTER 1 THE INTEGERSEvery good
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28 CHAPTER 1 THE INTEGERSThe Euclid
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30 CHAPTER 1 THE INTEGERSTheorem 1.
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32 CHAPTER 1 THE INTEGERS9. Use ind
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34 CHAPTER 1 THE INTEGERSProgrammin
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36 CHAPTER 2 GROUPSZ n . Consider t
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38 CHAPTER 2 GROUPSSymmetriesA symm
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40 CHAPTER 2 GROUPSTable 2.2. Symme
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42 CHAPTER 2 GROUPSand 4. We define
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44 CHAPTER 2 GROUPSis the inverse o
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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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Page 354 and 355: 20Finite FieldsFinite fields appear
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Page 372 and 373: 21Galois TheoryA classic problem of
Page 374 and 375: 366 CHAPTER 21 GALOIS THEORYthe Gal
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Page 382 and 383: 374 CHAPTER 21 GALOIS THEORY{id, σ
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Page 396 and 397: 388 NOTATIONSymbol Description Page
Page 398 and 399: 390 NOTATIONSymbol Description Page
Page 400 and 401: 392 HINTS AND SOLUTIONSConversely,
Page 402 and 403: 394 HINTS AND SOLUTIONS4. (a)(c)( 1
Page 404 and 405: 396 HINTS AND SOLUTIONS7. (a) (0000
Page 406 and 407: 398 HINTS AND SOLUTIONS9. For any h
Page 408 and 409: 400 HINTS AND SOLUTIONS2. The four
Page 410 and 411: 402 HINTS AND SOLUTIONSChapter 17.
Page 412 and 413: 404 HINTS AND SOLUTIONSChapter 19.
Page 414 and 415: GNU Free Documentation LicenseVersi
Page 416 and 417: 408 GFDL LICENSEappear in the title
Page 418 and 419: 410 GFDL LICENSEH. Include an unalt
Page 420 and 421: 412 GFDL LICENSEply to the other wo
Page 422 and 423: IndexG-equivalent, 205G-set, 203nth
Page 424 and 425: 416 INDEXEquivalence relation, 15Eu
Page 426 and 427: 418 INDEXKlein, Felix, 46, 170, 245
Page 428: 420 INDEXnormalizer of, 222proper,
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