Math 361, Spring 2014, Assignment 7

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

1. Theorem characterizing constructible numbers in terms of certain towers of field extensions (we stated this only in class; it is not in the book).
2. Theorem characterizing $\mathrm{deg}(\mathrm{irr}(\alpha,\mathbb{Q}))$ whenever $\alpha$ is a constructible number.

Solve the following problems:

1. Section 32, problems 2(a-f) and 10.
2. Prove the quadratic formula. Specifically: let $F$ be any field with $\mathrm{char}\ F\neq2$, and let $p = ax^2 + bx + c$ be any polynomial in $F[x]$ of degree 2. Show that, if there is an element $\delta\in F$ with $\delta^2=b^2-4ac$, then $p$ splits over $F$ and has roots $\frac{-b\pm\delta}{2a}$. Also show that if there is no such $\delta$, then the quotient ring $E=F[x]/\left\langle x^2 - (b^2 - 4ac)\right\rangle$ is a field with $[E:F]=2$. Letting $\delta$ denote the coset of $x$, show that $p$ splits over $E$, again with roots $\frac{-b\pm\delta}{2a}$.
3. Prove that $\cos(20^{\circ})$ is not a constructible number. (Hint: use elementary trigonometric identities to prove that, for any angle $\theta$, one has $\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$. Conclude that $\cos(20^{\circ})$ is a root of the polynomial $p = 8x^3 - 6x - 1$. Next, use the Rational Root Theorem to show that $p$ is irreducible over $\mathbb{Q}$, and is hence a constant multiple of the minimal polynomial of $\cos(20^{\circ})$. The rest is up to you.)
4. Prove that the regular nonagon is not constructible with compass and straightedge.
5. Prove that the regular pentagon and the regular decagon are constructible with compass and straightedge, using the following steps:

(a) Draw a unit circle diagram showing that $\sin(144^{\circ}) = \sin(36^{\circ})$.

(b) Use trigonometric identities to show that, for any angle $\theta$, we have $\sin(4\theta) = 4\sin(\theta)\cos(\theta)(2\cos^2(\theta)-1)$.

(c) Show that $\cos(36^{\circ})$ is a root of the polynomial $p=8x^3-4x-1$.

(d) Show that $p$ admits the factorization $p=(2x+1)(4x^2-2x-1)$.

(e) Show that $\cos(36^{\circ})$ is a root of the polynomial $4x^2-2x-1$.

(f) Use the quadratic formula together with some geometric reasoning to show that $\cos(36^{\circ}) = \frac{1+\sqrt{5}}{4}$.

(g) Use more trigonometric identities and some elementary algebra to show that $\cos(72^{\circ}) = \frac{\sqrt{5}-1}{4}$.

(h) Locate an actual compass and straightedge, and construct a regular pentagon and a regular decagon. (Obviously this is not the quiz question, but it is extremely fun. For bonus points, take sticks and rope into a muddy field and lay out a pentagonal fence.)