Transcription of Galois Theory - math.berkeley.edu
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Galois Theory - math.berkeley.edu
Galois TheoryDavid CorwinAugust 19, 20090 PreliminariesRemark (Notation).|G|denotes the order of a finite groupG. [E:F]denotes the degree of a field extensionE/F. We writeH Gto mean thatHis a subgroup ofG, andNEGto mean thatNis a normal subgroup ofG. IfE/FandK/Fare two field extensions, then when we say thatK/Fiscontained inE/F, we mean via a homomorphism that assume the following basic facts in this set of notes, in addition to ele-mentary number Theory , group and ring Theory , and linear algebra:Fact a field, thenF[x] is a PID, so all nonzero prime ideals aremaximal and are generated by a single irreducible polynomial. This irreduciblepolynomial is the polynomial of lowest positive degree in the ideal and is uniquemodulo is an algebraic element in an extensionE/F, then the set of poly-nomials inF[x] which vanish at is a prime ideal generated by an irreduciblepolynomialf(x) (we can make this canonical by requiringf(x) to be monic),andF( ) =F[x]/(f(x)).
Galois Theory David Corwin August 19, 2009 0 Preliminaries Remark 0.1 (Notation). jGjdenotes the order of a nite group G. [E: F] denotes the degree of a eld extension E=F. We write H Gto mean that H is a subgroup of G, and NE Gto mean that N is a normal subgroup of G. If E=F and K=F are two eld extensions, then when we say that K=F is
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