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262 CHAPTER 15 POLYNOMIALSm ≥ n. Thenf(x) − g(x)[q(x) − (b m /a n )x m−n ] = f(x) − g(x)q(x)+ (b m /a n )x m−n g(x)= r(x) + (b m /a n )x m−n g(x)= r(x) + b m x m+ terms of lower degreeis in S. This is a polynomial of lower degree than r(x), which contradictsthe fact that r(x) is a polynomial of smallest degree in S; hence, deg r(x)
15.2 THE DIVISION ALGORITHM 263Corollary 15.5 Let F be a field. An element α ∈ F is a zero of p(x) ∈ F [x]if and only if x − α is a factor of p(x) in F [x].Proof. Suppose that α ∈ F and p(α) = 0. By the division algorithm, thereexist polynomials q(x) and r(x) such thatp(x) = (x − α)q(x) + r(x)and the degree of r(x) must be less than the degree of x − α. Since thedegree of r(x) is less than 1, r(x) = a for a ∈ F ; therefore,p(x) = (x − α)q(x) + a.But0 = p(α) = 0 · q(x) + a = a;consequently, p(x) = (x − α)q(x), and x − α is a factor of p(x).Conversely, suppose that x−α is a factor of p(x); say p(x) = (x−α)q(x).Then p(α) = 0 · q(x) = 0.□Corollary 15.6 Let F be a field. A nonzero polynomial p(x) of degree n inF [x] can have at most n distinct zeros in F .Proof. We will use induction on the degree of p(x). If deg p(x) = 0, thenp(x) is a constant polynomial and has no zeros. Let deg p(x) = 1. Thenp(x) = ax + b for some a and b in F . If α 1 and α 2 are zeros of p(x), thenaα 1 + b = aα 2 + b or α 1 = α 2 .Now assume that deg p(x) > 1. If p(x) does not have a zero in F , then weare done. On the other hand, if α is a zero of p(x), then p(x) = (x − α)q(x)for some q(x) ∈ F [x] by Corollary 15.5. The degree of q(x) is n − 1 byProposition 15.2. Let β be some other zero of p(x) that is distinct from α.Then p(β) = (β − α)q(β) = 0. Since α ≠ β and F is a field, q(β) = 0. Byour induction hypothesis, p(x) can have at most n − 1 zeros in F that aredistinct from α. Therefore, p(x) has at most n distinct zeros in F . □Let F be a field. A monic polynomial d(x) is a greatest commondivisor of polynomials p(x), q(x) ∈ F [x] if d(x) evenly divides both p(x)and q(x); and, if for any other polynomial d ′ (x) dividing both p(x) and q(x),d ′ (x) | d(x). We write d(x) = gcd(p(x), q(x)). Two polynomials p(x) andq(x) are relatively prime if gcd(p(x), q(x)) = 1.
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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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Page 228 and 229: 13The Sylow TheoremsWe already know
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Page 240 and 241: 14RingsUp to this point we have stu
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Page 246 and 247: 238 CHAPTER 14 RINGSProposition 14.
Page 248 and 249: 240 CHAPTER 14 RINGSa ring homomorp
Page 250 and 251: 242 CHAPTER 14 RINGSTheorem 14.9 Le
Page 252 and 253: 244 CHAPTER 14 RINGSTheorem 14.15 L
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Page 260 and 261: 252 CHAPTER 14 RINGS11. Prove that
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Page 264 and 265: 15PolynomialsMost people are fairly
Page 266 and 267: 258 CHAPTER 15 POLYNOMIALSExample 1
Page 268 and 269: 260 CHAPTER 15 POLYNOMIALSProof. Su
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Page 302 and 303: 17Lattices and BooleanAlgebrasThe a
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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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