Intro_To_Galois_Theory

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1、Course311:HilaryTerm2000PartIII:IntroductiontoGaloisTheoryD.R.WilkinsContents3IntroductiontoGaloisTheory23.1RingsandFields.........................23.2Ideals...............................43.3QuotientRingsandHomomorphisms..............53.4TheCharacteristi

2、cofaRing...................73.5PolynomialRings.........................73.6Gauss'sLemma..........................103.7Eisenstein'sIrreducibilityCriterion...............123.8FieldExtensionsandtheTowerLaw..............123.9AlgebraicFieldExtensions..........

3、..........143.10RulerandCompassConstructions................163.11SplittingFields..........................213.12NormalExtensions........................243.13Separability............................253.14FiniteFields............................273.15TheP

4、rimitiveElementTheorem.................303.16TheGaloisGroupofaFieldExtension.............313.17TheGaloiscorrespondence....................333.18QuadraticPolynomials......................353.19CubicPolynomials........................353.20QuarticPolynomial

5、s.......................363.21TheGaloisgroupofthepolynomialx42...........373.22TheGaloisgroupofapolynomial................393.23SolvablepolynomialsandtheirGaloisgroups..........393.24Aquinticpolynomialthatisnotsolvablebyradicals.....4313IntroductiontoGal

6、oisTheory3.1RingsandFieldsDenitionAringconsistsofasetRonwhicharedenedoperationsofadditionandmultiplicationsatisfyingthefollowingaxioms:x+y=y+xforallelementsxandyofR(i.e.,additioniscommutative);(x+y)+z=x+(y+z)forallelementsx,yandzofR(i.e.,additionisass

7、ociative);thereexistsananelement0ofR(knownasthezeroelement)withthepropertythatx+0=xforallelementsxofR;givenanyelementxofR,thereexistsanelementxofRwiththepropertythatx+(x)=0;x(yz)=(xy)zforallelementsx,yandzofR(i.e.,multiplicationisassociative);x(y+z)

8、=xy+xzand(x+y)z=xz+yzforallelementsx,yandzofR(theDistributiveLaw).Lemma3.1LetRbearing.Thenx0=0and0x=0forallelementsxofR.ProofThezeroelement0ofRsatises0+0=0.UsingtheDistributiveLaw,wededucethatx0+x0=x(0+0)=x0and0x+0