ahw7 - Math 5126 Monday, March 17 Seventh Homework...

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Math 5126 Monday, March 17 Seventh Homework Solutions 1. Section 14.2, Exercise 2 on page 581. Determine the minimal polynomial over Q for the element 1 + 3 2 + 3 4. Let α denote the above element. We find a Galois extension K containing α , and then consider the conjugates containing α under the action of the Galois group Gal ( K / Q ) . Thus here we take K = Q ( 3 2 , ω ) , where ω is a primitive cube root of 1. Then Gal ( K / Q ) is generated by complex conjugation and the automorphism determined by 3 2 7→ ω ( 3 2 ) and ω 7→ ω . Thus the conjugates of α under the Galois group are 1 + 3 2 + 3 4 , 1 + ω 3 2 + ω 2 3 4 , 1 + ω 2 3 2 + ω 3 4 . This will yield a polynomial of degree 3. Since there are 3 conjugates under Gal ( K / Q ) , the minimal polynomial must have degree at least 3, and thus the polynomial will be the minimal polynomial. Explicitly, the polynomial is ( x - 1 - 3 2 - 3 4 )( x - 1 - ω 3 2 - ω 2 3 4 )( x - 1 - ω 2 3 2 - ω 3 4 ) = x 3 - 3 x 2 - 3 x - 1 .
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This note was uploaded on 01/29/2009 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.

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ahw7 - Math 5126 Monday, March 17 Seventh Homework...

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