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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 nitedimensional over K , it is generated over K by nitely many algebraic elements 1 ;:::; s . It follows from part (a) that L is the splitting eld of f D f 1 f 2 f s , where f i is the minimal polynomial of i over K . n Recall that a nitedimensional eld 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 nitedimensional eld extension K L , the following are equivalent: (a) The extension is Galois. (b) The extension is separable, and for all 2 L the minimal poly-nomial of over K splits into linear factors over L . (c) L is the splitting eld of a separable polynomial in Kx ....
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- Fall '08