HW5_322_F09

HW5_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 #5 Week #5: October 19th - October 25th. Topics : Rings, fields, ideals, quotient rings, Chinese remainder theorem, integral domains, the field of fractions, PIDs, UFDs, Euclidean domains, polynomials, power series, Laurent series, Eisensteins criterion. Read : FGT [1-6] (also a good idea to check out Langs Algebra , Ch. II) Problems due Friday, October 23rd, at 4:00 pm in Martin Luus mailbox: Problem 1 : Let F 4 be a field with four elements { , 1 ,x,y } . Write down the tables for addition and multiplication, hence showing F 4 is unique. Problem 2 : Let D n be the dihedral group of order 2 n , where n > 2 . Pick generators r and s such that r n = e , s 2 = e , and rs = sr- 1 . Write n as 2 m k , where k is odd. For each i m , show that Z i ( D n ) is generated by r n/ 2 i . Moreover, verify that Z i ( D n ) = Z m ( D n ) for every i m ....
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This note was uploaded on 05/07/2010 for the course MAT 322 at Princeton.

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HW5_322_F09 - Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY...

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