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276 CHAPTER 15 POLYNOMIALS(a) x 4 − x 2 − 3x + 2 = 0(b) x 4 + x 3 − 7x 2 − x + 6 = 0(c) x 4 − 2x 2 + 4x − 3 = 0(d) x 4 − 4x 3 + 3x 2 − 5x + 2 = 0
16Integral DomainsOne of the most important rings we study is the ring of integers. It was ourfirst example of an algebraic structure: the first polynomial ring that weexamined was Z[x]. We also know that the integers sit naturally inside thefield of rational numbers, Q. The ring of integers is the model for all integraldomains. In this chapter we will examine integral domains in general, answeringquestions about the ideal structure of integral domains, polynomialrings over integral domains, and whether or not an integral domain can beembedded in a field.16.1 Fields of FractionsEvery field is also an integral domain; however, there are many integraldomains that are not fields. For example, the integers Z are an integraldomain but not a field. A question that naturally arises is how we mightassociate an integral domain with a field. There is a natural way to constructthe rationals Q from the integers: the rationals can be represented as formalquotients of two integers. The rational numbers are certainly a field. In fact,it can be shown that the rationals are the smallest field that contains theintegers. Given an integral domain D, our question now becomes how toconstruct a smallest field F containing D. We will do this in the same wayas we constructed the rationals from the integers.An element p/q ∈ Q is the quotient of two integers p and q; however,different pairs of integers can represent the same rational number. For instance,1/2 = 2/4 = 3/6. We know thatab = c dif and only if ad = bc. A more formal way of considering this problemis to examine fractions in terms of equivalence relations. We can think of277
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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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Page 240 and 241: 14RingsUp to this point we have stu
Page 242 and 243: 234 CHAPTER 14 RINGSExample 3. We c
Page 244 and 245: 236 CHAPTER 14 RINGS3. (−a)(−b)
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
Page 254 and 255: 246 CHAPTER 14 RINGSthe Senate is n
Page 256 and 257: 248 CHAPTER 14 RINGShas a solution.
Page 258 and 259: 250 CHAPTER 14 RINGSMultiplying the
Page 260 and 261: 252 CHAPTER 14 RINGS11. Prove that
Page 262 and 263: 254 CHAPTER 14 RINGS34. Let R be a
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
Page 270 and 271: 262 CHAPTER 15 POLYNOMIALSm ≥ n.
Page 272 and 273: 264 CHAPTER 15 POLYNOMIALSPropositi
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Page 282 and 283: 274 CHAPTER 15 POLYNOMIALSAdditiona
Page 286 and 287: 278 CHAPTER 16 INTEGRAL DOMAINSelem
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Page 292 and 293: 284 CHAPTER 16 INTEGRAL DOMAINSCons
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Page 302 and 303: 17Lattices and BooleanAlgebrasThe a
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Page 320 and 321: 18Vector SpacesIn a physical system
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Page 330 and 331: 19FieldsIt is natural to ask whethe
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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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