Unformatted text preview: p.x/ splits into linear factors over L , and the roots of p.x/ are simple. This proves parts (a) and (b). Since L is ﬁnite–dimensional over K , it is generated over K by ﬁnitely many algebraic elements ˛ 1 ;:::;˛ s . It follows from part (a) that L is the splitting ﬁeld of f D f 1 f 2 ³³³ f s , where f i is the minimal polynomial of ˛ i over K . n Recall that a ﬁnite–dimensional ﬁeld extension K ± L is said to be Galois if Fix . Aut K .L// D K . Combining the last results gives the following: Theorem 9.4.14. For a ﬁnite–dimensional ﬁeld extension K ± L , the following are equivalent: (a) The extension is Galois. (b) The extension is separable, and for all ˛ 2 L the minimal polynomial of ˛ over K splits into linear factors over L . (c) L is the splitting ﬁeld of a separable polynomial in KŒxŁ ....
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 Fall '08
 EVERAGE
 Algebra, Polynomials, 2 L, minimal polynomial, Galois

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