Class Field Towers

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1、ClassFieldTowersFranzLemmermeyerSeptember7,2010Contents1SomeHistory11.1ThePrehistory..........................................11.1.1Fermat,Euler,andGauß:BinaryQuadraticForms..................11.1.2KummerandDedekind:Ideals.............................2

2、1.1.3KummerandWeber:KummerTheory.........................31.2TheGenesisoftheHilbertClassField.............................51.2.1KroneckerandWeber:ComplexMultiplication....................51.2.2Hilbert:HilbertClassFields..............................61

3、.2.3Furtw¨angler:SingularNumbers.............................61.2.4Takagi:ClassFieldTheory...............................71.2.5FromHilberttoHasse..................................81.3GenusClassFields........................................91.3.1Quadra

4、ticNumberFields................................91.3.2AbelianNumberFields.................................101.3.3GeneralNumberFields.................................101.3.4CentralExtensions....................................111.42-Classfields..........

5、.................................151.4.1QuadraticNumberFields................................151.4.2CubicNumberFields..................................181.4.3GeneralNumberFields.................................191.53-ClassFields.......................

6、....................191.5.1QuadraticNumberFields................................191.5.2CubicFields.......................................211.6`-ClassFields...........................................221.6.1QuadraticNumberFields.........................

7、.......221.6.2CyclicNumberFields..................................221.7Separants.............................................221.8CapitulationofIdealClasses..................................231.8.1Hilbert’sTheorem94..................................23

8、1.8.2Artin’sReduction....................................241.8.3ScholzandTaussky...................................251.8.4PrincipalIdealTheorems................................301.9ClassFieldTowers........................................311.9.