HW8_322_F09

# HW8_322_F09 - Fall 2009 MAT 322 ALGEBRA WITH GALOIS THEORY...

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Unformatted text preview: Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY PROBLEM SET #8 Week #9: November 23rd - November 29th. (Thanksgiving week) Topics : Perfect fields, Galois extensions, field automorphisms, the Galois group. Read : FGT [22-28] Problems due Friday, December 4th, at 4:00 pm in Martin Luu’s mailbox: • Problem 1 : Let F be a field of characteristic p > , and fix an algebraic closure ¯ F . For each k > , show that the following subset is a subfield, F p- k = { x ∈ ¯ F : x p k ∈ F } ⊂ ¯ F. Let F per be the union of all these subfields. Show that F per is the smallest perfect subfield of ¯ F , containing F . That is, the perfect closure of F in ¯ F . • Problem 2 : Let E/F be a finite extension, and assume E is a perfect field. Show that F is perfect. Is this true in general for algebraic extensions? • Problem 3 : Show that ( F per : F ) is infinite if F is imperfect. • Problem 4 : Let F be a field, and fix an algebraic closure ¯ F . Let F sep be the set of all α ∈ ¯ F such that Irr...
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HW8_322_F09 - Fall 2009 MAT 322 ALGEBRA WITH GALOIS THEORY...

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