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\title{Cheat Sheet, section ?}
\author{Jason Wojciechowski}

\begin{document}

\section{Definitions}

\begin{enumerate}
\item A simple extension field $K(\alpha):K$ is called
  \underline{algebraic} if $\alpha$ is the zero of a polynomial in
  $K[x]$.
  
\item Otherwise, a simple extension field $K(\alpha):K$ is called
  \underline{transcendental}.
  
\item The \underline{degree} of a simple extension $F(\alpha):F$ is
  the degree of the minimum polynomial of $\alpha$ over $F$.

\item A vector space $V$ over a field $K$ is a set $V$ such that

  \begin{enumerate}
    
  \item $V$ is a group under $+$.
    
  \item $k \cdot v \in V$ if $k \in K, v \in V$.
    
  \item $(k \times l) \cdot v = k \cdot (l \cdot v), k, l \in K, v \in V$.
      
  \item $(k+l)\cdot v = k\cdot v + l\cdot v, k,l\in K, v \in V$.

  \item $k(u+v) = k\cdot u + k \cdot v, k\in K, u,v \in V$.

  \item $1v = v, 1$ is unity, $v \in V$.
  
  \end{enumerate}
    
\item A \underline{basis} is a linearly independent spanning set.

\item A \underline{spanning set} is a set such that every vector in
  the vector space is a linear combination of the elements of the
  spanning set.
  
\item The \underline{dimension} of a vector space is the number of
  elements in its basis.

\item The dimension of $E$ over $F$ is denoted $[E:F]$.
  
\item If $\alpha$ is algebraic over $K$, then $[K(\alpha):K]$ is the
  degree of the minimum polynomial of $\alpha$.
\end{enumerate}

\section{Theorem}

\begin{thm}
  If $K$ is a field, then there is only one transcendental simple
  extension (up to isomorphism).
\end{thm}
  
\begin{thm} 
  If $K(\alpha)$ is an algebraic extension, then there exists a unique
  monic irreducible polynomial of minimum degree, $p(x)$, such that
  $\alpha$ is a zero of $p(x)$.
\end{thm}
  
\begin{cor}
  If $m(\alpha)=0$, then the minimum polynomial $p(x)$ will divide
  $m$.
\end{cor}
  
\begin{thm} 
  If $K(\alpha)$ is an algebraic extension of $K$ and $p(x)$ is the
  minimum polynomial of $\alpha$, then any element in $K(\alpha)$ can
  be written as a polynomial in $\alpha$ of degree less than the
  degree of $p(x)$.
\end{thm}

\begin{thm} 
  If $\alpha$ is algebraic over $K$, then $K(\alpha) = K[\alpha]$.
\end{thm}
  
\begin{thm} 
  If $m(x)$, a polynomial of degree greater than $0$, is irreducible
  in $K[x]$, then $K[x]/<m(x)>$ is a field.
\end{thm}
  
\begin{thm} 
  If $K$ is a field and $m$ a monic irreducible polynomial, then there
  exists $K(\alpha):K$ such that $\alpha$ is a zero of $m$.  
\end{thm}

\begin{thm} 
  If $E$ is an extension of $F$, then $E$ is a vector space.
\end{thm}

\end{document}