Sectioned Lecture Notes

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\title{Algebra I: Full Semester Lecture Notes}
\author{Compiled by First Last}

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\lecture{1}{Groups and Subgroups -- Sept 3}
\begin{definition}
A group is a set $G$ with a binary operation satisfying associativity,
identity, and inverses.
\end{definition}

\begin{theorem}
The identity in a group is unique.
\end{theorem}
\begin{proof}
If $e, e'$ are identities, $e = e e' = e'$.
\end{proof}

\begin{exercise}
Prove that inverses are unique in any group.
\end{exercise}

\lecture{2}{Homomorphisms and Isomorphisms -- Sept 5}
A group homomorphism is a map respecting the operation.

\begin{theorem}[First Isomorphism Theorem]
If $\phi: G \to H$ is a surjective homomorphism with kernel $N$, then $G/N \cong H$.
\end{theorem}

\lecture{3}{Rings -- Sept 10}
\begin{definition}
A ring is an abelian group under addition with an associative multiplication
that distributes over addition.
\end{definition}

\begin{exercise}
Show that $\mathbb{Z}_n$ is a commutative ring.
\end{exercise}

\lecture{4}{Fields -- Sept 12}
A field is a commutative ring in which every nonzero element is a unit.

\begin{theorem}
Every finite field has order $p^n$ for some prime $p$.
\end{theorem}

\end{document}