please write the answer and logic clearly.

here is handout and hw.

hope nice writing.

MAT 150C: MODERN ALGEBRA

Homework 7

Instructions. Please write the answer to each problem, including the computational ones, in connected

sentences and explain your work. Just the answer (correct or not) is not enough. Write your name in every

page and upload to Gradescope with the correct orientation. Make sure to indicate to Gradescope which

pages correspond to each problem. Finally, if you used another sources or discussed the problem

with classmates, be sure to acknowledge it in your homework.

Recall that an angle θ is constructible if and only if | cos(θ)| is a constructible real number.

1. (a) Show that the angle 60◦ is constructible.

√

√

(b) Prove (using either Euler’s identity e −1θ = cos(θ) + −1 sin(θ) or the multiplication law in the

group SO2 ) that cos(3θ) = 4 cos3 (θ) − 3 cos(θ). Note: If you use an identity for the sine or cosine

of the sum of two angles, you must prove it.

(c) Reduce modulo 5 to show that the polynomial 8×3 − 6x − 1 is irreducible in Q[x]. Using (b),

conclude that the angle 20◦ is not constructible. Thus, it is impossible to trisect an angle using

only ruler and compass.

2. Show that the angle θ = 15◦ is constructible by ruler and compass.

3. Show that the regular pentagon is constructible by ruler and compass.

4. Is the regular 9-gon constructible by ruler and compass?

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