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\begin{document}

\begin{center}
{\bf \Large \underline{Honors Algebra 4, MATH 371 Winter 2010}}\\ 
	{\large Assignment 6}\\Due Wednesday, March 24 at 08:35
\end{center}


\begin{enumerate}

	\item Let $K/F$ be a degree $2$ extension of fields.  
	\begin{enumerate}
		\item If the characteristic of $F$ is not 2, prove that $K=F(a)$ for some $a\in K\setminus F$ 
		with $a^2\in F$.\label{one}
		
		\item Give a counterexample to (\ref{one}) if $F$ has characteristic 2. 
	
		\item Fix $F$ of characteristic not 2 and let $K_1,K_2$ be quadratic extensions of $F$
		with $K_1 = F(a_1)$ and $K_2=F(a_2)$ where $a_i^2=b_i\in F$.  Prove that $K_1\simeq K_2$
		as extensions of $F$ (i.e. that there exists an isomorphism of fields $K_1\simeq K_2$
		restricting to the identity on $F$) if and only if $b_1/b_2\in (F^{\times})^2$ is a square.\label{three}
		Conclude that the isomorphism classes of quadratic extensions of $F$ are in bijection with
		the group $F^{\times}/(F^{\times})^2$.
		
		\item Using (\ref{three}), give a complete list (without repetition) of all isomorphism classes of quadratic
		extensions of $\Q$.
		
	\end{enumerate}

	\item For $a\in \F_p$, set
	$$f_a(x):=X^p-X-a\in \F_p[X].$$
	\begin{enumerate}
		\item If $a=0$, show that $f_a(X)=\prod_{u\in \F_p} (X-u)$.
		\item Suppose that $a\neq 0$ and let $E_a$ be a splitting field of $f_a(X)$.
		If $r_1,r_2\in E_a$ are roots of $f_a$, prove that $r_1-r_2\in \F_p$.
		\item Show that $f_a(X)$ is irreducible for all $a\in \F_p^{\times}$.
		\item Prove that $f_b(X)$ splits completely over $E_a$ for each fixed $a\in \F_p^{\times}$
		and all $b\in \F_p^{\times}$.  Conclude that $E_a$ is independent of $a$.
	\end{enumerate}


	\item Find the minimal polynomials of $2\cos(2\pi/5)$ and $2\cos(2\pi/7)$ over $\Q$.

	\item For each of the following extensions, determine $[K:F]$ and an $F$-basis of $K$.
	\begin{enumerate}
		\item $F=\Q$, $L=\Q(a,b)$ with $a^2=6$ and $b^3=2$.
		
		\item $F=\C(T)$ and $L$ is the splitting field of $X^n-T$ over $F$, with $n$ a positive integer.
		
		\item $F=\F_p(T)$ and $L$ is the splitting field of $X^p-T$ over $F$, with $p$ a prime.
	\end{enumerate}
	
	\item Let $K/F$ be a finite extension of fields and let $\alpha\in K$.  Then $\alpha$ induces
	an $F$-linear map of finite-dimensional $F$-vector spaces
	$$m_{\alpha}:K\rightarrow K.$$
	\begin{enumerate}
		\item Prove that $\alpha$ is a root of the characteristic polynomial of the linear map $m_{\alpha}$.
		Hint: select a suitable $F$-basis of $F(\alpha)$. \label{charp}
		
		\item Use (\ref{charp}) to find a monic degree 3 polynomial with $\Q$-coefficients 
		satisfied by $1+\sqrt[3]{2}+\sqrt[3]{4}$.
		
		\item Prove that if $K=F(\alpha)$, then the characteristic polynomial of $m_{\alpha}$
		as a linear map $K\rightarrow K$ is in fact the minimal polynomial of $\alpha$ over $F$.
	\end{enumerate} 
	
	\item For each of the following algebraic elements $\alpha$ of the given field extension 
	$K/\Q$, express $1/\alpha$ and $1/(\alpha+1)$ as polynomials in $\alpha$ with $\Q$-coefficients.
	\begin{enumerate}
		\item $K$ is the splitting field of $f=X^3-3X+1$ and $\alpha$ is a root of $f$.
		
		\item $K$ is the splitting field of $f=X^4+X^3+x^2+x+1$ and $\alpha$ is a root of $f$.
	
		\item $K$ is the splitting field of $f=X^5-3X+3$ and $\alpha$ is a root of $f$.
	\end{enumerate}
	
	\item Prove that $X^4-5$ is irreducible over $\Q$ and has splitting field $K$ of degree $8$ over $\Q$.
	Describe this splitting field explicitly as $\Q(a,b)$ where $a$ is a root of $X^4-5$ and $b^2\in \Q$.
	In terms of $a$ and $b$, write down a $\Q$-basis for $K$.
	
	\item Describe the splitting fields of $f:=X^3-5$ over $\F_{11}$ and $\F_7$ and factor $f$
	into linear factors over each extension.




\end{enumerate}
\end{document}