Hw10 - p q p 2 √ 2 where p,q are of the form a b √ 2 with a,b rational Represent 1 p 2 √ 2 2-3 p 2 √ 2 in this form(5 Find a tower of

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(1) Let a be a constructible number. Show that a is also constructible. (2) (a) let x 0 be a root of ( 2+1) x 7 +(5 - 2 2) x 4 +( 2 - 2) = 0. Con- struct a polynomial f ( x ) of degree 14 with rational coeﬃcients such that f ( x 0 ) = 0. (b) Let F be a number ﬁeld and let r F . Suppose x 0 is a root of polynomial of n -degree with coeﬃcients in F ( r ). Show that it’s a root of a polynomial of degree 2n with coeﬃcients in F . (3) Explain how to construct 2+ 3 3 using ruler and compass. (4) Let F be the ﬁeld consisting of real numbers of the form
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Unformatted text preview: p + q p 2 + √ 2 where p,q are of the form a + b √ 2, with a,b rational. . Represent 1 + p 2 + √ 2 2-3 p 2 + √ 2 in this form. (5) Find a tower of ﬁelds Q = F ⊂ F 1 ⊂ F 2 ⊂ F 3 such that q 1 + √ 2 + √ 3 ∈ F 3 Show that all the steps in the tower except for the last one are nontrivial. I.e show that F 6 = F 1 , and F 1 6 = F 2 . 1...
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This note was uploaded on 04/26/2011 for the course MATH 246 taught by Professor Applebaugh during the Spring '10 term at University of Toronto- Toronto.

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