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HW10_322_F09

# HW10_322_F09 - Fall 2009 MAT 322 ALGEBRA WITH GALOIS THEORY...

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Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY PROBLEM SET #10 Week #11: December 7th - December 13th. Topics : Normal closure, compositum, translation principle, cyclotomic ﬁelds, ﬁnite ﬁelds, normal bases, cyclic extensions, norm and trace, Hilbert’s theorem 90, Artin-Schreier, Galois solvability theorem, geometric construction problems. Read : FGT [50-60] and [64-68] and [14-16] Problems due Friday, December 18th, at 4:00 pm in Martin Luu’s mailbox: Problem 1 : Let p be a prime. Write down a normal basis for Q ( ζ p ) over Q . Now let n be any positive integer. Is the following statement true or false: { ζ a n } a ( Z /n Z ) × is always a normal basis for Q ( ζ n ) over Q ? Problem 2 : Find an element β in F 8 df = F 2 [ x ] / ( x 3 + x + 1) such that the three elements { β, β 2 , β 4 } form a normal basis for F 8 over F 2 . Problem 3 : Use Hilbert’s Theorem 90 for Q ( i ) to solve the Pythagorean equation over the integers: Let x, y, z be nonzero integers satisfying

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HW10_322_F09 - Fall 2009 MAT 322 ALGEBRA WITH GALOIS THEORY...

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