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ProblemSet11 - Q x p x 6 Let F = Z 2 and let f x = x 3 x 1...

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Homework 11 Due: NEVER 1. Describe the elements of Q ( 3 5). Justify carefully using an appropriate theorem. 2. Show Q ( 2 , 3) = Q ( 2 + 3). 3. Find the splitting field for x 3 - 1 over Q . Express your answer in the form Q ( a ). 4. Find the splitting field for x 4 + 1 over Q . 5. Find a polynomial p ( x ) in Q [ x ] so that Q ( 1 + 5) is isomorphic to
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Unformatted text preview: Q [ x ] / ( p ( x )). 6. Let F = Z 2 and let f ( x ) = x 3 + x +1 ∈ F [ x ]. Suppose a is a zero of f ( x ) in some extension of F . How many elements does F ( a ) have? Express each member of F ( a ) in terms of a . Write out a complete multiplication table for F ( a )....
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