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Introduction to Abstract Algebra (Math 113)

IntroductiontoAbstractAlgebra(Math113) +and ,CosetsandLagrange (ProofsOmitted).. , , (ProofsOmitted).. ,themostlikelyresponsewillbe: Somethinghorribletodowithx,yandz .Ifyou reluckyenoughtobumpintoamathematicianthe nyoumightgetsomethingalongthelinesof: Algebraistheabstractencapsulationofourin tuitionforcomposition .Bycomposition, , , ,evenasalittlebaby. N:={1,2, }, +and . Z:={.. 2, 1,0,1,2,..}, +and .AdditiononZhasparticularlygoodpropertie s, 2Q:={ab|a,b Z,b =0}, +and .Thistime,multiplicationishasparticularl ygoodproperties, ,therealnumbersandthenC, +and , +and +b=b+aforalla,b Q,ora (b+c)=a b+a cforalla,b,c (setswithextrastructure),ofwhichZandQare definitivemembers.

5 Field Extensions and Galois Theory 76 ... Remarks. 1. Notice that the reflexive property implies that x ∈ [x]. Hence equivalence classes are non-empty and their union is S. 2. The symmetric and transitive properties imply that y ∈ [x] if and only if [y] = [x].

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