103b_06w_fs

103b_06w_fs - Math 103B Final Exam Solutions 23 March 2006...

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Unformatted text preview: Math 103B Final Exam Solutions 23 March 2006 1. Since it is a splitting field, the extension is Galois, so we can use the fundamental theorem of Galois theory for (a)–(c). (a) The degree of the extension is the order of the Galois group, which is 8. (b) This is equivalent to asking for the number of subgroups of order 2 of Z 2 ⊕ Z 4 . Each element of order 2 generates such a group. The elements of order 2 are (0 , 2), (1 , 0) and (1 , 2). Thus the answer is 3. (b) This is equivalent to [ E : K ] = 4 and so we need subgroups of order 4. They are { } + Z 4 , { (0 , 0) , (1 , 1) , (0 , 2) , (1 , 3) } , { (0 , 0) , (0 , 2) , (1 , 0) , (1 , 2) } , and so the answer is also 3. (c) The answer is zero since [ E : K ] must divide [ E : Q ] = 8. Alternatively the answer is zero since a group of order 8 cannot have a subgroup of order 3. 2. (a) Ideals are closed under multiplication by elements of R and by addition. Thus a i b i ∈ B and the sum of such terms is also in B . Likewise, they are in....
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This note was uploaded on 06/18/2009 for the course MATH 103b taught by Professor Staff during the Spring '08 term at UCSD.

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103b_06w_fs - Math 103B Final Exam Solutions 23 March 2006...

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