Mathematical Structures: Fields

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%%run pdflatex

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 \documentclass[12pt]{amsart}
 \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
 \parindent=0pt
 \parskip=5pt
 \addtolength{\oddsidemargin}{-.5in}
 \addtolength{\evensidemargin}{-.5in}
 \addtolength{\textwidth}{1in}
 \theoremstyle{definition}
 \newtheorem{definition}{Definition}
 \newtheorem*{morphisms}{Morphisms}
 \newtheorem*{basic_results}{Basic Results}
 \newtheorem*{examples}{Examples}
 \newtheorem{example}{}
 \newtheorem*{properties}{Properties}
 \newtheorem*{finite_members}{Finite Members}
 \newtheorem*{subclasses}{Subclasses}
 \newtheorem*{superclasses}{Superclasses}
 \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
 \pagestyle{myheadings}\thispagestyle{myheadings}
 \markboth{\today}{math.chapman.edu/structures}
 
 \begin{document}
 \textbf{\Large Fields}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Fields}{edit}
 
 \abbreviation{Fld}
 \begin{definition}
 A \emph{field} is a \href{Commutative_rings_with_identity.pdf}{commutative rings with identity} $\mathbf{F}=\langle F,+,-,0,\cdot,1
 \rangle$ such that
 
 
 $\mathbf{F}$ is non-trivial:  $0\ne 1$
 
 
 every non-zero element has a multiplicative inverse:  $x\ne 0\implies \exists y 
 (x\cdot y=1)$
 
 Remark: 
 The inverse of $x$ is unique, and is usually denoted by $x^{-1}$.
 
 \end{definition}
 
 \begin{morphisms}
 Let $\mathbf{F}$ and $\mathbf{G}$ be fields. A morphism from $\mathbf{F}$
 to $\mathbf{G}$ is a function $h:F\rightarrow G$ that is a homomorphism: 
 
 $h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
 
 Remark: 
 It follows that $h(0)=0$ and $h(-x)=-h(x)$.
 
 \end{morphisms}
 \begin{basic_results}
 $0$ is a zero for $\cdot$: $0\cdot x=x$ and $x\cdot 0=0$.
 \end{basic_results}
 \begin{examples}
 \begin{example}
 $\langle\mathbb{Q},+,-,0,\cdot,1\rangle$, the field of rational numbers with addition, subtraction, zero, multiplication, and one.
 \end{example}
 \end{examples}
 \begin{table}[h]
 \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})
 
 \begin{tabular}{|ll|}\hline
 Classtype & first-order\\\hline
 Equational theory & \\\hline
 Quasiequational theory & \\\hline
 First-order theory & \\\hline
 Locally finite & no\\\hline
 Residual size & unbounded\\\hline
 Congruence distributive & yes\\\hline
 Congruence modular & yes\\\hline
 Congruence n-permutable & yes, $n=2$\\\hline
 Congruence regular & yes\\\hline
 Congruence uniform & yes\\\hline
 Congruence extension property & \\\hline
 Definable principal congruences & \\\hline
 Equationally def. pr. cong. & \\\hline
 Amalgamation property & \\\hline
 Strong amalgamation property & \\\hline
 Epimorphisms are surjective & \\\hline
 \end{tabular}
 \end{properties}
 \end{table}
 \begin{finite_members} $f(n)=$ number of members of size $n$.
 
 $\begin{array}{lr}
 f(1)= &0\\
 f(2)= &1\\
 f(3)= &1\\
 f(4)= &1\\
 f(5)= &1\\
 f(6)= &0\\
 There exists one field, called the Galois field $GF(p^m)$ of each prime-power order $p^m$.
 \end{array}$
 \end{finite_members}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Fields_of_characteristic_zero.pdf}{Fields of characteristic zero} 
 
 \href{Algebraically_closed_fields.pdf}{Algebraically closed fields} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Integral_domains.pdf}{Integral domains} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
 %

%%run pdflatex

%

 
 \documentclass[12pt]{amsart}
 \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
 \parindent=0pt
 \parskip=5pt
 \addtolength{\oddsidemargin}{-.5in}
 \addtolength{\evensidemargin}{-.5in}
 \addtolength{\textwidth}{1in}
 \theoremstyle{definition}
 \newtheorem{definition}{Definition}
 \newtheorem*{morphisms}{Morphisms}
 \newtheorem*{basic_results}{Basic Results}
 \newtheorem*{examples}{Examples}
 \newtheorem{example}{}
 \newtheorem*{properties}{Properties}
 \newtheorem*{finite_members}{Finite Members}
 \newtheorem*{subclasses}{Subclasses}
 \newtheorem*{superclasses}{Superclasses}
 \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
 \pagestyle{myheadings}\thispagestyle{myheadings}
 \markboth{\today}{math.chapman.edu/structures}
 
 \begin{document}
 \textbf{\Large Fields}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Fields}{edit}
 
 \abbreviation{Fld}
 \begin{definition}
 A \emph{field} is a \href{Commutative_rings_with_identity.pdf}{commutative rings with identity} $\mathbf{F}=\langle F,+,-,0,\cdot,1
 \rangle$ such that
 
 
 $\mathbf{F}$ is non-trivial:  $0\ne 1$
 
 
 every non-zero element has a multiplicative inverse:  $x\ne 0\implies \exists y 
 (x\cdot y=1)$
 
 Remark: 
 The inverse of $x$ is unique, and is usually denoted by $x^{-1}$.
 
 \end{definition}
 
 \begin{morphisms}
 Let $\mathbf{F}$ and $\mathbf{G}$ be fields. A morphism from $\mathbf{F}$
 to $\mathbf{G}$ is a function $h:F\rightarrow G$ that is a homomorphism: 
 
 $h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
 
 Remark: 
 It follows that $h(0)=0$ and $h(-x)=-h(x)$.
 
 \end{morphisms}
 \begin{basic_results}
 $0$ is a zero for $\cdot$: $0\cdot x=x$ and $x\cdot 0=0$.
 \end{basic_results}
 \begin{examples}
 \begin{example}
 $\langle\mathbb{Q},+,-,0,\cdot,1\rangle$, the field of rational numbers with addition, subtraction, zero, multiplication, and one.
 \end{example}
 \end{examples}
 \begin{table}[h]
 \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})
 
 \begin{tabular}{|ll|}\hline
 Classtype & first-order\\\hline
 Equational theory & \\\hline
 Quasiequational theory & \\\hline
 First-order theory & \\\hline
 Locally finite & no\\\hline
 Residual size & unbounded\\\hline
 Congruence distributive & yes\\\hline
 Congruence modular & yes\\\hline
 Congruence n-permutable & yes, $n=2$\\\hline
 Congruence regular & yes\\\hline
 Congruence uniform & yes\\\hline
 Congruence extension property & \\\hline
 Definable principal congruences & \\\hline
 Equationally def. pr. cong. & \\\hline
 Amalgamation property & \\\hline
 Strong amalgamation property & \\\hline
 Epimorphisms are surjective & \\\hline
 \end{tabular}
 \end{properties}
 \end{table}
 \begin{finite_members} $f(n)=$ number of members of size $n$.
 
 $\begin{array}{lr}
 f(1)= &0\\
 f(2)= &1\\
 f(3)= &1\\
 f(4)= &1\\
 f(5)= &1\\
 f(6)= &0\\
 \end{array}$
 
 There exists one field, called the Galois field $GF(p^m)$ of each prime-power order $p^m$.
 \end{finite_members}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Fields_of_characteristic_zero.pdf}{Fields of characteristic zero} 
 
 \href{Algebraically_closed_fields.pdf}{Algebraically closed fields} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Integral_domains.pdf}{Integral domains} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
 %

http://mathcs.chapman.edu/structuresold/files/Fields.pdf 
%%run pdflatex


%

\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\addtolength{\oddsidemargin}{-.5in}
\addtolength{\evensidemargin}{-.5in}
\addtolength{\textwidth}{1in}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\pagestyle{myheadings}\thispagestyle{myheadings}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Fields}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Fields}{edit}

\abbreviation{Fld}
\begin{definition}
A \emph{field} is a \href{Commutative_rings_with_identity.pdf}{commutative rings with identity} $\mathbf{F}=\langle F,+,-,0,\cdot,1
\rangle$ such that


$\mathbf{F}$ is non-trivial:  $0\ne 1$


every non-zero element has a multiplicative inverse:  $x\ne 0\implies \exists y 
(x\cdot y=1)$

Remark: 
The inverse of $x$ is unique, and is usually denoted by $x^{-1}$.

\end{definition}

\begin{morphisms}
Let $\mathbf{F}$ and $\mathbf{G}$ be fields. A morphism from $\mathbf{F}$
to $\mathbf{G}$ is a function $h:F\rightarrow G$ that is a homomorphism: 

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

Remark: 
It follows that $h(0)=0$ and $h(-x)=-h(x)$.

\end{morphisms}
\begin{basic_results}
$0$ is a zero for $\cdot$: $0\cdot x=x$ and $x\cdot 0=0$.
\end{basic_results}
\begin{examples}
\begin{example}
$\langle\mathbb{Q},+,-,0,\cdot,1\rangle$, the field of rational numbers with addition, subtraction, zero, multiplication, and one.
\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & first-order\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence extension property & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr}
f(1)= &0\\
f(2)= &1\\
f(3)= &1\\
f(4)= &1\\
f(5)= &1\\
f(6)= &0\\
\end{array}$

There exists one field, called the Galois field $GF(p^m)$ of each prime-power order $p^m$.
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\ 

\href{Fields_of_characteristic_zero.pdf}{Fields of characteristic zero} 

\href{Algebraically_closed_fields.pdf}{Algebraically closed fields} 

\end{subclasses}
\begin{superclasses}\ 

\href{Integral_domains.pdf}{Integral domains} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%







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Last edited July 8, 2004 2:28 pm by Jipsen (diff)
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