Tutorial 1 : Algebraic Extensions and Ruler and Compass Constructions

1. Let $ K/F$ be an extension of degree $ n.$
    (a) For any $ a \in K,$ prove that the map $ \mu_a: K \rightarrow K$ defined by $ \mu_a(x)=ax$ for all $ x \in K,$ is a linear transformation of the $ F$-vector space $ K.$ Show that $ K$ is isomorphic to a subfield of the ring $ F^{n \times n}$ of $ n \times n$ matrices with entries in $ F.$
    (b) Prove that $ a$ is a root of the characteristic polynomial of $ \mu_a.$ Use this procedure to find monic polynomials satisfied by $ \sqrt[3]{2}$ and $ 1+\sqrt[3]{2}+\sqrt[3]{4}.$

2. Prove that $ -1$ is not a sum of squares in the field $ \mathbb{Q}(\beta)$ where $ \beta = \sqrt[3]{2}\; e^{2 \pi i/3}.$

3. Let $ k$ be a field and $ x$ be an indeterminate. Find the irreducible polynomial of $ x$ over $ k(y)$ where $ y=x^3/(x+1).$

4. Find an algebraic extension $ K$ of $ \mathbb{Q}(x)$ such that the polynomial $ f(y)=y^2 - x^3/(x^2+1) \in \mathbb{Q}(x)[y]$ has a root in $ K.$

5. The construction of a regular $ 7$-gon amounts to the construction of the real number $ \alpha=\cos(2\pi/7).$ Show that $ \alpha$ is a root of $ f(x)=x^3+x^2-2x-1.$ Hence conclude that a regular $ 7$-gon is not constructible by ruler and compass.

6. Show that an angle of $ n$ degrees, $ n \in \mathbb{N},$ is constructible if and only if $ 3 \mid n.$

7. Prove that it is impossible, in general, to quintsect an arbitrary angle by ruler an compass. Is it possible to divide the angle $ 60$ degrees into five equal parts by ruler and compass ?