\part{Fundamentals}

Mathematical logic
Set theory
Linear algebra
Group theory
Elementary number theory
Ring theory




\chapter{Field extensions}



Fields are algebraic that generalize arithmetic on $\mathbb{Q},\mathbb{R}$.

Since fields are a special class of rings, many fundamental properties of fields are uncovered naturally in the study of ring theory.



\section{Field extension}

$E$ is a field extension of $F$ iff $F$ is a subfield of $E$
$S$ is a field extension of $R$ iff $R$ is a subring of $S$

\section{Algebraic extension}

Special type of field extension
$\mathbb{R}(i)$
$\mathbb{Q}(\sqrt{2})$



\chapter{Algebraic equations}


\begin{definition}[Algebraic equation]
An \emph{algebraic equation of $F$} is an equation of the following form, defined by the valuation of some polynomial $P \in F[X]$.
\[ P(x) = 0\]
\end{definition}