MAT 150C Algebra Questions

MAT 150C: MODERN ALGEBRAHomework 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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