College Algebra Exam Review 422

# College Algebra Exam Review 422 - p.x splits into linear...

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432 9. FIELD EXTENSIONS – SECOND LOOK The following is the converse to the previous proposition: Proposition 9.4.13. Suppose K ± L is a ﬁeld extension, dim K .L/ is ﬁnite, and Fix . Aut K .L// D K . (a) For any ˇ 2 L , ˇ is algebraic and separable over K , and the minimal polynomial for ˇ over K splits in LŒxŁ . (b) For ˇ 2 L , let ˇ D ˇ 1 ;:::;ˇ n be a list of the distinct elements of f ±.ˇ/ W ± 2 Aut K .L/ g . Then .x ² ˇ 1 /.:::/.x ² ˇ n / is the minimal polynomial for ˇ over K . (c) L is the splitting ﬁeld of a separable polynomial in KŒxŁ . Proof. Since dim K .L/ is ﬁnite, L is algebraic over K . Let ˇ 2 L , and let ˇ D ˇ 1 ;:::;ˇ r be the distinct elements of f ±.ˇ/ W ± 2 Aut K .L/ g . Deﬁne g.x/ D .x ² ˇ 1 /.:::/.x ² ˇ r / 2 LŒxŁ . Ev- ery ± 2 Aut K .L/ leaves g.x/ invariant, so the coefﬁcients of g.x/ lie in Fix . Aut K .L/ D K . Let p.x/ denote the minimal polynomial of ˇ over K . Since ˇ is a root of g.x/ , it follows that p.x/ divides g.x/ . On the other hand, every root of g.x/ is of the form ±.ˇ/ for ± 2 Aut K .L/ and, therefore, is also a root of p.x/ . Since the roots of g.x/ are simple, it follows that g.x/ divides p.x/ . Hence p.x/ D g.x/ , as both are monic. In particular,
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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 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 poly-nomial 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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