Mathematical Structures: Fields

# Fields

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http://mathcs.chapman.edu/structuresold/files/Fields.pdf
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\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
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\newtheorem*{finite_members}{Finite Members}
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\begin{document}
\textbf{\Large Fields}

\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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