1.
Show that it is impossible to construct a regular \(9\)-gon with straightedge and compass.
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Worksheet Tutorial-4
Following exercises are taken from Sections 13.3 and 13.4 of the book Abstract Algebra by Dummit and Foote.
2.
Use the fact that \(\alpha=2\cos(2\pi/5)\) satisfies the equation \(x^2+x-1=0\) to construct a regular pentagon with straightedge and compass.
3.
Determine the splittting field and degree over \(\Q\) of the following polynomials.
\begin{equation*}
x^4-2;\quad x^4+2;\quad x^4+x^2+1;\quad\text{and}\quad x^6-4
\end{equation*}
4.
Let \(K/F\) be a finite field extension. Show that \(K\) is a splitting field over \(F\) if and only if every irreducible polynomial in \(F[x]\) that has a root in \(K\) splits completely in \(K[x]\text{.}\)
5.
Let \(K_1,K_2\) be finite extensions of a field \(F\) contained in a field \(K\text{.}\) Furthermore, assume that \(K_1,K_2\) are splitting fields over \(F\text{.}\) Show that \(K_1K_2\) and \(K_1\cap K_2\) are splitting fields over \(F\text{.}\)
