Math 361, Spring 2017, Assignment 12

Carefully define the following terms, then give one example and one non-example of each:

1. Splitting field (of a non-constant polynomial $f\in F[x]$).
2. Root tower (in an extension $F\rightarrow E$).
3. Solvable by radicals.

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning trisection of angles.
2. Theorem concerning existence of splitting fields.
3. Theorem concerning uniqueness of splitting fields.

Solve the following problems:

(Construction of the regular pentagon)

1. Prove that $\cos\left(72^{\circ}\right) = \frac{\sqrt{5}-1}{4}.$ (Hint: put $\theta=72^{\circ}$. Draw a picture showing why $\cos(3\theta)=\cos(2\theta)$. Manipulate this equation using various trigonometric identities, eventually showing that $\cos\left(72^{\circ}\right)$ is a root of the polynomial $4x^3-2x^2-3x+1$. Give a geometric argument for why $\cos\left(72^{\circ}\right)\neq1$, then divide this polynomial by $x-1$. Finally, use the quadratic formula.)
2. Devise a compass-and-straightedge construction that produces a regular pentagon.
3. Prove that the regular nonagon (nine-sided polygon) cannot be constructed with compass and straightedge.
4. Construct a splitting field for $f=x^4-x\in\mathbb{Z}_2[x]$, then factor $f$ completely over your field. What polynomial function does $f$ induce on your field?
5. Is the equation $x^4-x=0$ solvable by radicals over $\mathbb{Z}_2$? Why or why not?

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