Math 361, Spring 2016, Assignment 10

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

1. Constructible number.
2. Splitting field (of some non-constant polynomial $f\in F[x]$).

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

1. Theorem concerning the degrees of constructible numbers.
2. Theorem concerning the impossibility of three classically-sought geometric constructions.
3. Theorem concerning the existence and uniqueness of splitting fields.

Solve the following problems:

1. Section 32, problem 10.
2. Show that the regular pentagon can be constructed with compass and straightedge. (Hint: it suffices to show that $\cos(72^\circ)$ is a constructible number. Use trigonometric identities to express $\cos(3\theta)$ and $\cos(2\theta)$ in terms of $\cos(\theta)$ and $\sin(\theta)$, and explain why for $\theta=72^\circ$ these expressions must be equal. Then use the resulting relation to calculate $\cos(72^\circ)$ exactly. Finally, you can appeal to Theorem 32.6, or better still, devise an explicit construction of the pentagon.)
3. Construct the splitting field of $x^3-1\in\mathbb{Q}[x]$. Specifically, find a $\mathbb{Q}$-basis for the splitting field, and carry out several sample calculations. Then do the same for $x^4-1$ and, if you want a challenge, $x^5-1$. (These fields are called the third, fourth, and fifth cyclotomic fields, respectively. Cyclotomic fields appear in many number-theoretic applications and have been intensively studied. The word "cyclotomic" means "circle-cutting;" do you see why this name was chosen for these fields?)

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