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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 23 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.
 Spring '10
 Applebaugh
 Math

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