Notes on \emph{"A book of Abstract Algebra - by Charles C. Pinter"}, is a $LaTeX$ version of handmade notes taken while reading the book. It contains only some definitions and theorems (without proofs), so it is highly recommended to read the actual book instead of the current notes. Additionally, some theorems and concepts are extended with notes from other resources from outside the book.
Notes on \emph{"A book of Abstract Algebra - by Charles C. Pinter"}, is a $LaTeX$ version of handmade notes taken while reading the book. It contains only some definitions and theorems (without proofs), so it is highly recommended to read the actual book instead of the current notes.
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\emph{This is an unfinished and 'work in progress' document.}
\end{abstract}
@ -324,16 +324,114 @@ for any elements a, b, and c in the ring if $a \neq 0$.
A ring has the \emph{cancellation property} iff it has no \emph{divisors of zero}.
\end{theorem}
\begin{definition}[Ideal]
A nonempty subset $B$ of a ring $A$ is called an \emph{ideal} of $A$ if $B$ is closed with respect to addition and negatives, and $B$ absorbs products in $A$.
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(\emph{Absorbs product}: $\forall b \in B$ and $x \in A$, then $xb, bx \in B$).
\end{definition}
\begin{definition}[Principal ideal]
A \emph{principal ideal} is an ideal $I$ in a ring $R$ that is generated by a single element $a \in R$ through multiplication by every element of $R$. In other words $I = aR =\{a r : r \in R \}$.
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(eg. Every ideal of $\mathbb{Z}$ is principal).
\end{definition}
\begin{definition}[Integral domain]
An \emph{integral domain} is defined to be a commutative ring with unity having the cancellation property.
\end{definition}
Every field is an integral domain, but the converse is not true (eg. $\mathbb{Z}$ is an integral domain but not a field).
\begin{definition}[Ideal]
A nonempty subset $B$ of a ring $A$ is called an \emph{ideal} of $A$ if $B$ is closed with respect to addition and negatives, and $B$ absorbs products in $A$.
\begin{definition}[Characteristic n]
Let $A$ be a ring with unity, the \emph{characteristic} of $A$ is the least positive integer $n$ such that
$$1+1+\cdots+1=0$$
If there is no such positive integer $n$, $A$ has characteristic $0$.
\end{definition}
\section{Factoring into primes}
\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 $p|n$.
\end{definition}
\framebox{WIP: covered until chapter 18, work in progress.}
\begin{theorem}[Factorization into primes]
Ever integer $n>1$ can be expressed as a product of positive primes. That is, there are one or more primes $p_1, \ldots, p_r$ such that $n=p_1 p_2\cdots p_r$.
\end{theorem}
\begin{theorem}[Unique factorization]
Suppose $n$ can be factored into positive primes in two ways, namely,
$$n= p_1\cdots p_r = q_1\cdots q_t$$
Then $r=t$, and the $p_i$ are the same numbers as the $q_j$ except, possibly, for the order in which they appear.
\end{theorem}
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,
$$a^{p-1}\equiv1\pmod p, \forall a \not\equiv0\pmod p$$
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So, by taking $a^{p-2}\cdot a \equiv1\pmod p$, where $a^{p-2}\equiv a^{-1}\pmod p$ (the inverse modulo p), we see that $a^p \equiv a \pmod p, \forall a \in\mathbb{Z}$, so $a^p - a$ is a multiple of $p$.
\end{theorem}
\begin{theorem}[Euler's $\phi$ function]
\emph{Euler's $\phi$ function} describes the number of integers in $\mathbb{Z}/ n \mathbb{Z}$ which are relatively prime (coprime) to $n$.
\end{theorem}
\begin{theorem}[Euler's theorem]
If $a$ and $n$ are relatively prime, $a^{\phi(n)}\equiv1\pmod n$.
\end{theorem}
\section{Polynomials}
\begin{definition}
Let $A$ be a commutative ring with unity, and $x$ an arbitrary symbol. Every expression of the form
$$a_0+ a_1 x + a_2 x^2+\cdots+ a_n x^n$$
is called a \emph{polynomial in $x$ with coefficients in $A$}, or more simply, a \emph{polynomial in $x$ over $A$}.
\end{definition}
The expressions $a_k x^k$, for $k \in\{1, \ldots, n \}$, are called the \emph{terms} of the polynomial, being $a_n x^n$ the \emph{leading term}, and $a_0$ the \emph{constant term}.
The $a_k$ are called the \emph{coefficients} of $x^k$, being $a_n$ the \emph{leading coefficient}. And the \emph{degree} of a polynomial $a(x)$ is the greatest $n$ such that the coefficient of $x^n$ is not zero.
The polynomial whose leading coefficient is equal to $1$ is called \emph{monic}.
\begin{theorem}[Division algorithm for polynomials]
If $a(x)$ and $b(x)$ are polynomials over a field $F$, and $b(x)\neq0$, there exist polynomials $q(x)$ and $r(x)$ over $F$ such that
$a(x)= b(x) q(x)+ r(x)$ and
[$r(x)=0$ or $\deg r(x) < \deg b(x)$].
\end{theorem}
\begin{theorem}
Any two nonzero polynomials $a(x), b(x)\in F[x]$ have a $\gcd d(x)$. Furthermore, $d(x)$ can be expressed as a \emph{linear combination}
$$d(x)= r(x) a(x)+ s(x) b(x)$$
where $r(x), s(x)\in F[x]$.
\end{theorem}
\begin{theorem}[Factorization into irreducible polynomials]
Every polynomial $a(x)$ of positive degree in $F[x]$ can be written as a product
$$a(x)= k p_1(x) p_2(x)\cdots p_r(x)$$
where $k$ is a constant in $F$ and $p_1(x), \ldots, p_r(x)$ are monic irreducible polynomials of $F[x]$.
\end{theorem}
\begin{theorem}[Unique factorization]
If $a(x)$ can be written in two ways as a product of monic irreducibles, say
$$a(x)= k p_1(x)\cdots p_r(x)= l q_1(x)\cdots q_s(x)$$
then $k=l$, $r=s$, and $p_i(x)= q_j(x)$.
\end{theorem}
\begin{theorem}
$c$ is a root of $a(x)$ iff $x - c$ is a factor of $a(x)$.
\end{theorem}
\begin{theorem}
If $a(x)$ has distinct roots $c_1, \ldots, c_m$ in $F$, then $(x-c_1)(x-c_2)\cdots(x-c_m)$ is a factor of $a(x)$.
\end{theorem}
\begin{theorem}
If $a(x)$ has degree $n$, it has at most $n$ roots.
\end{theorem}
\framebox{WIP: covered until chapter 26, work in progress.}
