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Question

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

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2

(1) The Golden Ratio is the number φ = 1+  5 . 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 equa- tion has no constructible solutions:

x3 - 6x + 2√2 = 0.

Hint: You can use Theorem 12.3.22 if you make an appropriate sub- stitution.

(4)  Textbook section 12.5, problem 23: Let t be a transcendental num- ber.  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 =  2 .  Is    θ a constructible

3           3

angle? Prove your claim.

√3    √

(6) Is . √2

2  7

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.