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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 workdont 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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