HW7_322_F09 - Fall 2009 MAT 322 ALGEBRA WITH GALOIS THEORY...

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Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY PROBLEM SET #7 Week #8: November 16th - November 22nd. Topics : Algebraic extensions, transcendental elements (ex Liouville numbers), algebraically closed fields, algebraic closure, Zorn’s lemma, splitting fields, normal extensions, separable extensions, separable closure, separable degree. Read : FGT [11-13], [17-21], and [69-72] Problems due Friday, November 20th, at 4:00 pm in Martin Luu’s mailbox: Problem 1 : Let n = p m 1 1 · · · p m t t be the prime factorization. Prove that Φ n ( x ) = Φ p 1 ··· p t ( x p m 1 - 1 1 ··· p m t - 1 t ) . Problem 2 : Let E be an algebraic extension of F , and let σ : E E be an F -linear embedding of E into itself. Show that σ is an automorphism. Problem 3 : Let E be an algebraic extension of F . Assume every f F [ x ] splits completely into a product of linear factors over E . Show that E is then algebraically closed. That is, the above is true for every f E [ x ] .
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