HW9_322_F09

HW9_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 #9 Week #10: November 30th - December 6th. Topics : Dedekinds inequality, finite subgroups of F are cyclic, primitive element theorem, Artins theorem, Galois criteria, the Galois correspondence, ex x 3- 2. Read : FGT [29-34] and [36-42] Problems due Friday, December 11th, at 4:00 pm in Martin Luus mailbox: Problem 1 : Let E be the splitting field of x 4- 4 x 2 + 2 Q [ x ] . Describe the automorphisms of E , and find a generator for the group Gal( E/ Q ) . Problem 2 : Find a primitive element (over Q ) for each subfield of Q ( 13 ) , and draw a diagram showing all inclusions among these subfields, and find their relative degrees. Do the same for Q ( 17 ) . What is noteworthy? Problem 3 : For an odd prime p , we let ( p ) denote the Legendre symbol. That is, it is the unique non-trivial group homomorphism p : ( Z /p Z ) { 1 } , a 7 a p ....
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This note was uploaded on 05/07/2010 for the course MAT 322 at Princeton.

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HW9_322_F09 - Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY...

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