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Math 461 F
Spring 2011
Homework 3
Drew Armstrong
Reading.
Section 2.4
Problems.
A.1.
How many diﬀerent (complex) numbers does the expression
p
1 +
√
3
represent? Find a polynomial over
Z
which has these numbers as its roots.
A.2.
Use the trigonometric identity cos(3
θ
) = 4 cos
3
θ

3 cos
θ
together with
Cardano’s formula to ﬁnd an expression for cos(
π/
9). (Note: This expression
must
involve complex numbers because cos(
π/
9) is not constructible.)
A.3.
Suppose that
p
= 2
a
+ 1 is a prime number. Show that
a
must be a
power of 2. (Hint: If
a
has an
odd
factor
b
, show that the polynomial
x
b
+1
factors nicely.)
A.4.
Prove that
√
2 = 1 +
1
2 +
1
2 +
1
2 +
.
.
.
.
(You can assume that the expression on the right converges.) We can de
scribe this process
recursively
by setting
s
0
= 1 and
s
n
= 1 + 1
/
(1 +
s
n

1
)
for
n
≥
1. What is
s
4
? How close is this to
√
2?
Let
D
be the set of numbers that can be formed from
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This note was uploaded on 01/08/2012 for the course MATH 561 taught by Professor Armstrong during the Spring '11 term at University of Miami.
 Spring '11
 Armstrong
 Math, Algebra

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