Transcription of GALOIS THEORY - University of Washington
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GALOIS THEORY - University of Washington
GALOIS THEORYWe will assume on this handout that is an algebraically closed field. This means thatevery irreducible polynomial in [x] is of degree 1. Suppose thatFis a subfield of andthatKis a finite extension ofFcontained in . For example, we can take =C, the fieldof complex :We will say that Kis a normal extension ofF if the following statementis true:(1)If is any embedding ofKinto overF, then (K) = following statements are equivalent to(1).(2)There exists a polynomialf(x) F[x] such thatKis the splitting field forf(x) in overF.(3)If K, then the minimal polynomialp(x) for overFfactors as a product of linearpolynomials inK[x].The basic propositions of GALOIS TheoryWe will make the following assumptions in all of the propositions on this handout withoutfurther A:The fieldFis either a field of characteristic zero or a finite B:The fieldKis a finite, normal extension about assumptionAis satisfied, then normal extensions ofFare also called GALOIS extensions ofF.
GALOIS THEORY We will assume on this handout that is an algebraically closed eld. This means that every irreducible polynomial in [x] is of degree 1. Suppose that F is a sub eld of and that Kis a nite extension of Fcontained in . For example, we can take = C, the eld
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