Galois Theory - Dartmouth College

1 Galois TheoryMath 31 Summer 2013 Dartmouth CollegeAugust 21, 2013 Galois and Abel Evariste GaloisNiels Henrik AbelMath 31 Summer 2013 Galois TheoryThe Idea of GaloisPermute the roots off(x) = (x2 2)(x2 3): = 27 2 27 2 37 3 37 3 1= 27 2 27 2 37 3 37 3 2= 27 2 27 2 37 3 37 3 3= 27 2 27 2 37 3 37 3 Math 31 Summer 2013 Galois TheoryModern Galois TheoryPermutation of roots = change of basis for splitting fieldK{1, 2, 3, 6} 3 {1, 2, 3, 6}This map is a ring homomorphism fromKtoK, and it s actuallyan automorphism ofKthat 31 Summer 2013 Galois TheoryModern Galois TheoryThese automorphisms are denoted by:Gal(K/Q) ={ Aut(K) : (a) =afor alla Q}. Galois ,Gal(K/Q) =Gal(f).Note:|Gal(K/Q)|= [K:Q].Math 31 Summer 2013 Galois TheorySubgroup LatticeEx:f(x) = (x2 2)(x2 3),K=Q( 2, 3){ ,(1 2),(3 4),(1 2)(3 4)}kkkkkkkkkkkkkkkSSSSSSSSSSSSSSS{ ,(1 2)}SSSSSSSSSSSSSSSSS{ ,(1 2)(3 4)}{ ,(3 4)}kkkkkkkkkkkkkkkkk{ }Math 31 Summer 2013 Galois TheorySubgroups and FieldsH1={ ,(3 4)}fixes the fieldQ( 2).

2 H2={ ,(1 2)}fixes the fieldQ( 3).H3={ ,(1 2)(3 4)}fixes the fieldQ( 6).KuuuuuuuuuuIIIIIIIIIIGal(K/Q)uuuuuuuu uuIIIIIIIIIIQ( 2)Q( 3)Q( 6)H1H2H3 QIIIIIIIIII uuuuuuuuuu{ }IIIIIIIIII uuuuuuuuuuMath 31 Summer 2013 Galois TheorySubgroups and FieldsK2uuuuuuuuuu22 IIIIIIIIII{ }2uuuuuuuuuu22 IIIIIIIIIIQ( 2)Q( 3)Q( 6)H1H2H3Q2 IIIIIIIIII22uuuuuuuuuuGal(K/Q)2 IIIIIIIIII22uuuuuuuuuuIn general: ifFis a subfield ofK, then[F:Q] = [Gal(K/Q) : Gal(K/F)].Math 31 Summer 2013 Galois TheoryGalois Theory in a NutshellThe main things to take away:1|Gal(K/Q)|= [K:Q]2 There is a one-to-one correspondence between subgroups ofGand subfields a subfield ofK, then[F:Q] = [Gal(K/Q) : Gal(K/F)].Math 31 Summer 2013 Galois Theory