103b_06w_fe

# 103b_06w_fe - Math 103B Final Exam(80 points 23 March 2006...

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Unformatted text preview: Math 103B Final Exam (80 points) 23 March 2006 • Please put your name and ID number on your blue book. • The exam is CLOSED BOOK except for both sides of two sheets of notes. • Calculators are NOT allowed. • In a multipart problem, you can do later parts without doing earlier ones. • You must show your work to receive credit. 1. (20 pts.) There is a polynomial f ( x ) ∈ Q [ x ] such that its splitting field E over Q has Gal( E/ Q ) ≈ Z 2 ⊕ Z 4 . Remember to show your work—don’t just give numbers. (a) What is [ E : Q ] and why? (b) How many fields K ⊂ E are there such that [ E : K ] = 2? (c) How many fields K ⊂ E are there such that [ K : Q ] = 2? (d) How many fields K ⊂ E are there such that [ E : K ] = 3? 2. (15 pts.) If A and B are ideals in a commutative ring R , then AB is the set of all sums a 1 b 1 + a 2 b 2 + ··· + a n b n where a i ∈ A , b i ∈ B and n > 0. It can be shown that AB is an ideal of R ....
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103b_06w_fe - Math 103B Final Exam(80 points 23 March 2006...

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