\centerline{Exercises VII}

\bigskip\noindent
{\bf 1.} Construct subfields of ${\bf C}$ which are splitting fields of
the following polynomial over ${\bf Q}$:
$$
X^3 - 1,\ 
X^4 + 5X^2 + 2,\ 
X^6 - 8.
$$
What are their degrees over ${\bf Q}$?

\medskip\noindent
{\bf 2.} Construct the splitting fields over ${\bf F}_3$ for the following
polynomials:
$$
X^3 + 2X + 1,\ 
X^3 + X^2 + X + 2.
$$
Are these fields isomorphic?

\medskip\noindent
{\bf 3.} Which of the following extensions are normal?
$$
{\bf Q}(X)/{\bf Q},\ 
{\bf Q}(\sqrt{-5})/{\bf Q},\ 
{\bf Q}(\alpha)/{\bf Q},\ 
{\bf Q}(\alpha, \sqrt{5})/{\bf Q}(\alpha).
$$
Here $\alpha$ is a real seventh root of $5$.

\medskip\noindent
{\bf 4.} Construct the normal closures of the following extensions:
$$
{\bf Q}(\alpha)/{\bf Q},\ 
{\bf Q}(\beta)/{\bf Q},\ 
{\bf Q}(\sqrt{2}, \sqrt{3})/{\bf Q},\ 
{\bf Q}(\gamma, \sqrt{2})/{\bf Q},\ 
{\bf Q}(\delta)/{\bf Q}.
$$
Here $\alpha^5 = 3$, $\beta^7=2$, $\gamma^3 = 2$ are real solutions and
$\delta$ is a root of $X^3 - 3X^2 + 3$.

\medskip\noindent
{\bf 5.} Find the Galois groups over ${\bf Q}$ of the extensions you
constructed in exercise 4.

\bye
