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\lecture{9: Field Extensions}{Week 9}
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\epigraph{In any field, find the strangest thing and then explore it.}{John Archibald Wheeler, physicist}

\section{Field Extensions}


We ended our constructible numbers lectures with the following classical questions:
\begin{itemize}
\item \textbf{Doubling the Cube}:  Can you construct a number $x$ such that the volume of the cube with side length $x$ is $2$?  In other words, can you construct $\sqrt[3]{2}$?
\item \textbf{Trisecting the Angle}:  Given any two lines $L, M$ that intersect at a unique point $P$ in the plane, can you always draw a third line $N$ through $P$ such that the angle between $N, L$ is a third of that between $M, L$?  

To give an explicit example: we can make a line that makes an angle of $\pi/3 = 60^\circ$ with the origin by constructing a circle with radius 2 around the origin, drawing a line perpendicular to the $x$-axis through $(1,0)$, finding their intersection $P$, and drawing the line through the origin and $P$.
\begin{center}
\includegraphics[width=4in]{compass_trisect.pdf}
\end{center}
Can you draw a line that makes an angle of $\pi/9 = 20^\circ$ with the origin?  In other words, can you construct $\cos(20^\circ)$?
\item \textbf{Squaring the Circle}:  Given a circle $C$ with radius 1, can you construct a point $P$ such that the distance from $P$ to the origin is the same as the circumference of $C$?  In other words, can you construct $\pi$?
\end{itemize}

We answer these questions in these notes.  To do this, we will need the following three tools:
\begin{enumerate}
\item The concept of \textbf{dimension}, as covered in our earlier notes.
\item The following theorem on constructible numbers, that we proved in week 8:
\begin{thm}
Let $a$ be any constructible value.  Then there are constructible values $b,c$ such that $a$ is a root of the polynomial
\begin{align*}
x^2 + bx + c.
\end{align*}
\end{thm}
\item The concept of a \textbf{field extension}.
\end{enumerate}

This third concept is not one that we have discussed yet!  We define it here:
\subsection{Field extensions:\ definitions, examples.}

\begin{defn}
(will be filled in when time allows!)
\end{defn}









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