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z$`xOW-3Bzb(806C$_7i9<|&Up;R6z1iwo68$nq3BUchW=Y)-$`7Z&R1qP=`1o*u1H zM=i<-)fg5R`4GE|b8vaOdCjS3$D^G0oQ)ZBH8b`0PZctquW!w&R_Z>kn_6#rBd=6+ zEcOOBH1&V>tJ&*uW8GJ}LIjsJ=zaNePm~Z`|bf?`s%plK&S?&H=Fi diff --git a/abstract-algebra-charles-pinter-notes.tex b/abstract-algebra-charles-pinter-notes.tex index ddd2e25..a3f5257 100644 --- a/abstract-algebra-charles-pinter-notes.tex +++ b/abstract-algebra-charles-pinter-notes.tex @@ -16,6 +16,9 @@ \newtheorem{definition}{Def}[section] \newtheorem{theorem}[definition]{Thm} +% ndiv command, to allow ΜΈ| (not-div) without needing extra packages +\newcommand{\ndiv}{\hspace{-4pt}\not|\hspace{2pt}} +\newcommand{\crossover}{\hspace{-0pt}\not\hspace{-2pt}} \title{Notes on "A book of Abstract Algebra", Charles C. Pinter} \author{arnaucube} @@ -237,9 +240,10 @@ Example of an \emph{homomorphism}: $f: \mathbb{Z}_6 \rightarrow \mathbb{Z}_3$. In an abelian group, every subgroup is normal. \end{definition} -\begin{definition}[Kernel] +\begin{definition}[Kernel]\label{def:homomorphismkernel} Let $f: G \rightarrow H$ be a homomorphism. The \emph{kernel} of $f$ is the set $K$ of all the elements of $G$ which are carried by $f$ onto the neutral element of $H$. That is, $$K = {x \in G : f(x) = e}$$ + \emph{Kernel in the context of Extension fields: \ref{def:extensionkernel}} \end{definition} For every homomorphism, the $e \in G$ maps to $e \in H$, so the \emph{kernel} is never empty, it always contains the identity $e_G$, and if the kernel only contains the identity, then $f$ is one-to-one (injective). @@ -439,7 +443,99 @@ From the last two theorems: every integer $m$ can be factored into primes, and t If $a(x)$ has degree $n$, it has at most $n$ roots. \end{theorem} +In finite $F$, polynomial $\neq$ polynomial function. If $F$ is infinite, polynomial $=$ polynomial function. -\framebox{WIP: covered until chapter 26, work in progress.} +For every polynomial with rational coefficients, there is a polynomial with integer coefficients having the same roots. See: +$$a(x) = \frac{k_0}{l_0} + \frac{k_1}{l_1} x + \cdots + \frac{k_n}{l_n} x^n$$ +$$=\frac{1}{l_0 \cdots l_n} \cdot \underbrace{(k_0 l_1 \cdots l_n + k_1 l_0 l_2 \cdots l_n x + \cdots + k_n l_0 \cdots l_{n-1} x^n)}_{b(x)}$$ +$a(x)$ has rational coefficients, $b(x)$ has integer coefficients. $b(x)$ differs from $a(x)$ only by a constant factor ($\frac{1}{l_0\cdots l_n}$), so $a(x)$ and $b(x)$ have the same roots. + +$\Longrightarrow~~\forall~p(x) \in \mathbb{Q}[x]$, there is a $f(x) \in \mathbb{R}$ with the same roots (for every polynomial with rational coefficients, there is a polynomial with integer coefficients having the same roots). + +\begin{theorem} + If $s/t$ is a root of $a(x)$, then $s|a_0$ and $t|a_n$. +\end{theorem} + +\begin{theorem} + Suppose $a(x)$ can be factured as $a(x) = b(x)c(x)$, where $b(x), c(x)$ have rational coefficients. Then there are polynomials $B(x), C(x)$ with integer coefficients, which are constant multiples of $b(x)$ and $c(x)$ respectively, such that $a(x) = B(x)C(x)$. +\end{theorem} + +\begin{theorem}[Eisenstein's irreducibility criterion] + Let $a(x) = a_0 + a_1 x + \cdots + a_n x^n$ be a polynomial with integer coefficients. + + If there is prime $p$ such that $p | a_i, ~\forall i\in\{0, n-1\}$, and $p \ndiv a_n$ and $p^2 \ndiv a_0$, then $a(x)$ is irreducible over $\mathbb{Q}$. + % Suppose there is a prime number $p$ which divides every coefficient of $a(x)$ except the leading coefficient $a_n$; suppose $p$ does not divide $a_n$ and $p^2$ does not divide $a_0$. Then $a(x)$ is irreducible over $\mathbb{Q}$. +\end{theorem} + +\section{Extensions of fields} + +\begin{definition}[Kernel]\label{def:extensionkernel} + The \emph{kernel} of $\sigma_c$ consists of all polynomials $a(x) \in F[x]$ such that $c$ is a root of $a(x)$. + + \emph{Kernel in the context of Homomorphisms: \ref{def:homomorphismkernel}} +\end{definition} + +\begin{definition}[Algebraic] + $c \in E$ is called \emph{algebraic over} $F$ if it is the root of some nonzero polynomial $a(x) \in F[x]$. + + Otherwise, $c$ is called \emph{transcendental over} $F$. +\end{definition} + +$E/K$ denotes the (field) extension of $E$ over $K$. + +\begin{theorem}[Basic theorem of field extensions] + Let $F$ be a field and $a(x) \in F[x]$ a nonconstant polynomial. There exists an extension field $E/F$ and an element $c \in E$ such that $c$ is a root of $a(x)$. +\end{theorem} + +Let $a(x) \in F[x]$ be a polynomial of degree $n$. There is an extension field $E/F$ which contains all $n$ roots of $a(x)$. + +\section{Vector spaces} + +\begin{definition}[Vector space] + A \emph{vector space} over a field $F$ is a set $V$, with two operations $+, \cdot$, called \emph{vector addition} and \emph{scalar multiplication}, such that + + \begin{itemize} + \item $V$ with vector addition is an abelian group + \item $\forall k \in F$ and $\overrightarrow{a} \in V$, the scalar product $k \overrightarrow{a}$ is an element of $V$, subject to the following conditions: + $\forall k, l \in F,~\overrightarrow{a},\overrightarrow{b} \in V$ + \begin{enumerate}[i.] + \item $k(\overrightarrow{a} + \overrightarrow{b}) = k\overrightarrow{a} + k\overrightarrow{b}$ + \item $(k + l)\overrightarrow{a} = k\overrightarrow{a} + k\overrightarrow{b}$ + \item $k(l\overrightarrow{a}) = (kl)\overrightarrow{a}$ + \item $1 \overrightarrow{a} = \overrightarrow{a}$ + \end{enumerate} + \end{itemize} +\end{definition} + +\begin{definition}[Linear combination] + If $\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n} \in V$, and $k_1, k_2, \ldots, k_n$ are scalars, then the vector + $$k_1 \overrightarrow{a_1} + k_2 \overrightarrow{a_2} + \cdots + k_n \overrightarrow{a_n}$$ + is called a \emph{linear combination} of $\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}$. + + The set of all the linear combinations of $\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}$ is a \emph{subspace of} $V$. +\end{definition} + +\begin{definition}[Linear dependancy] + Let $S = \{$\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}$\}$ be a set of distinct vectors in a vector space $V$. $S$ is said to be \emph{linearly dependent} if there are scalars $k_1, \ldots, k_n$, not all zero, such that $k_1 \overrightarrow{a_1} + k_2 \overrightarrow{a_2} + \cdots + k_n \overrightarrow{a_n} = 0$. + Which is equivalent to saying that at least one of the vectors in $S$ is a linear combination of the others. + + If $S$ is not linearly dependent, then it is \emph{linearly independent}. $S$ is linearly independent iff $k_1 \overrightarrow{a_1} + k_2 \overrightarrow{a_2} + \cdots + k_n \overrightarrow{a_n}=0$ implies $k_1 = k_2 = \cdots = k_n =0$. + Which is equivalent to saying thatno vector in $S$ is equal to a linear combination of the other vectors in $S$. +\end{definition} + +If $\{ \overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n} \}$ is linearly dependent, then some $a_i$ is a linear combination of the preceding ones. + +If $\{ \overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n} \}$ spans $V$, and $a_i$ is a linear combination of the preceding vectors, then $\{ \overrightarrow{a_1}, \ldots, \crossover{\overrightarrow{a_i}}, \ldots, \overrightarrow{a_n} \}$ still spans $V$. + + +\begin{theorem} + Any two bases of a vector space $V$ have the same number of elements. + + (This comes from the fact that all bases of $\mathbb{R}^n$ contain exactly $n$ vectors) +\end{theorem} + +If the set $(\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n})$ spans $V$, it contains a basis of $V$. + +\framebox{WIP: covered until chapter 28, work in progress.} \end{document}