Math 246F 2015: Homework 6-The Golden Ratio is the number φ

| August 30, 2017

Question
Math 246F 2015: Homework 6
Due at the beginning of tutorial on week 12
Wed, Dec 2, Thr Dec 3
Note: make sure to attach cover sheet!

Please turn in only problems 3–8. The rest of the problems are
suggested for practice.

(1) The Golden Ratio is the number φ =


1+ 5
2 .

Is φ constructible?

(2) Textbook section 12.5, problems 1, 4, 5, 7, 11, 12, 15, 18, 25.
(3) Textbook section 12.5, problem 16: Prove that the following equation has no constructible solutions:

x3 − 6x + 2 2 = 0.
Hint: You can use Theorem 12.3.22 if you make an appropriate substitution.
(4) Textbook section 12.5, problem 23: Let t be a transcendental number. Prove that t cannot be a root of any equation of the form
x2 + ax + b = 0, where a and b are constructible numbers. Hint: you
can use the fact that the constructible numbers are algebraic.
(5) Let θ be an acute angle. Suppose cos θ =
angle? Prove your claim.
(6) Is


2
√ √
3
2 7

2
3.

Is

θ
3

a constructible

constructible?



(7) Is Q( π) := {a + b π | a, b ∈ Q} a subfield of R?
(8) Give an example of a finite set which is extraordinary. Recall that
a set R is extraordinary if it contains itself: R ∈ R.

1

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