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\title{Math 421-621\\Spring 2013\\Homework 9}
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\centerline{\it Due Wednesday April 17, 2013.}

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\noindent 1. (5 pt) Let $\mathbb{F}\subset\mathbb{K}$ be fields such that $[\mathbb{K}:\mathbb{F}]$ is prime. Show that if $\xi$ is any element of $\mathbb{K}$ that is not in $\mathbb{F}$, then $\mathbb{K}=\mathbb{F}(\xi)$.

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\noindent 2. We say that the extension of fields $\mathbb{F}\subset\mathbb{K}$ is quadratic if $[\mathbb{K}:\mathbb{F}]=2$.
\begin{itemize}
\item[a)] (5 pt) Let $a,b,c\in\mathbb{F}$ with $a\neq 0$. Show that if $ax^2+bx+c$ is irreducible, then it has a solution in some quadratic extension in $\mathbb{F}$.
\item[b)] (5 pt) Derive the solution(s) of the equation $ax^2+bx+c=0$. You will need to make an assumtion, what is it?
\item[c)] (5 pt) Give an example of a field with infinitely many distinct quadratic extensions.
\item[d)] (5 pt) Give an example of a field with only one quadratic extension and an example of a field with no quadratic extensions.
\item[e)] (5 pt) Give an example of a field $\mathbb{F}$ and an irreducible (over $\mathbb{F}[x]$) quadratic polynomial $ax^2+bx+c$ that has only one root in $\mathbb{K}:=\mathbb{F}[x]/(ax^2+bx+c)$. 
\end{itemize}

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\noindent 3. Let $F_1$ and $F_2$ be fields both contained in a larger field $L$ and both containing $K$. We define the {\it composite} of the fields $F_1$ and $F_2$ ($F_1F_2$) to be the smallest subfield of $L$ containing both $F_1$ and $F_2$. 
\begin{itemize}
\item[a)] (5 pt) Show that $[F_1F_2:K]\leq [F_1:K][F_2:K]$.
\item[b)] (5 pt) Show that if $\text{gcd}([F_1:K],[F_2:K])=1$ then $[F_1F_2:K]=[F_1:K][F_2:K]$.
\end{itemize} 


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