Mathematical Structures: Fields

# Fields

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 %%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 $n$. $\begin{array}{lr} f(1)= &0\\ f(2)= &1\\ f(3)= &1\\ f(4)= &1\\ f(5)= &1\\ f(6)= &0\\ There exists one field, called the Galois field$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
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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 $n$.

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

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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