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arnaucube
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abstract-algebra-charles-pinter-notes.tex
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@@ -285,7 +285,7 @@ Quotient group construction is useful as a way of actually manufacturing all the
\section{Rings}
\begin{definition}[Ring]
A set $A$ with operations called \emph{addition} and \emph{multiplication} which satisfy the following axions:
A set $A$ with operations called \emph{addition} and \emph{multiplication} which satisfy the following axioms:
\begin{enumerate}[i.]
\item $A$ with addition alone is an abelian group.
\item Multiplication is associative.
@@ -348,7 +348,7 @@ Every field is an integral domain, but the converse is not true (eg. $\mathbb{Z}
If there is no such positive integer $n$, $A$ has characteristic $0$.
\end{definition}
\section{Factoring into primes}
\section{Elements of number theory}
\begin{definition}[Euclid's lemma]
Let $m$ and $n$ be integers, and let $p$ be a prime. If $p|(mn)$, then either $p|m$ or $p|n$.
@@ -366,9 +366,6 @@ Every field is an integral domain, but the converse is not true (eg. $\mathbb{Z}
From the last two theorems: every integer $m$ can be factored into primes, and the prime factors of $m$ are unique (except for the order).
\section{Elements of number theory}
\begin{theorem}[Little theorem of Fermat]
Let $p$ be a prime. Then,
$$a^{p-1} \equiv 1 \pmod p, \forall a \not\equiv 0 \pmod p$$
@@ -381,7 +378,7 @@ From the last two theorems: every integer $m$ can be factored into primes, and t
\end{theorem}
\begin{theorem}[Euler's theorem]
If $a$ and $n$ are relatively prime, $a^{\phi(n)} \equiv 1 \pmod n$.
If $a$ and $n$ are relatively prime, $$a^{\phi(n)} \equiv 1 \pmod n$$
\end{theorem}