\documentclass[12pt]{amsart}
\usepackage{amsmath,amsthm,amsfonts,latexsym,amssymb}
\pagestyle{headings}
\setlength{\textwidth}{36true pc}
\setlength{\headheight}{8true pt} % needed for 12pt documents
\setlength{\oddsidemargin}{15 truept}
\setlength{\evensidemargin}{15 truept}
\setlength{\textheight}{572true pt}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lm}[thm]{Lemma}
\newtheorem{prp}[thm]{Proposition}
\newtheorem{clm}[thm]{Claim}
%
\theoremstyle{definition}
\newtheorem{df}[thm]{Definition}
\newtheorem{exmp}[thm]{Example}
\newtheorem{prb}[thm]{Problem}
%
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
%
\numberwithin{equation}{section}
%
\newcommand{\thref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lmref}[1]{Lemma~\ref{#1}}
%
\DeclareMathOperator{\satf}{\models}
\DeclareMathOperator{\miff}{\Longleftrightarrow} 
                                           % math `if and only if'
\DeclareMathOperator{\mimplies}{\Longrightarrow} 
                                           % math `implies'
\DeclareMathOperator{\mimplied}{\Longleftarrow}  
                                           % math `implied by'
\newcommand{\set}[2]{\{#1\,:\,\text{#2}\}} % { x : ... }
\DeclareMathOperator{\id}{\textsl{id}}     % The identity map
\DeclareMathOperator{\miffdef}{\stackrel{def}{\miff}}
                                           % defining iff
\DeclareMathOperator{\eqdef}{\stackrel{def}{=}}
                                           % defining equation

\DeclareMathOperator{\neareq}{\stackrel{\ae}{=}}
\DeclareMathOperator{\aeqv}{\stackrel{ae}{\equiv}}
\DeclareMathOperator{\aeq}{\stackrel{ae}{=}}
%
\DeclareMathOperator{\BZ}{\mathbb{Z}} % integers
\DeclareMathOperator{\BQ}{\mathbb{Q}} % rationals
\DeclareMathOperator{\BR}{\mathbb{R}} % reals
\DeclareMathOperator{\BC}{\mathbb{C}} % complex numbers
\DeclareMathOperator{\BE}{\mathbb{E}} % 
\DeclareMathOperator{\BF}{\mathbb{F}} % 
\DeclareMathOperator{\BK}{\mathbb{K}} % 
\DeclareMathOperator{\BL}{\mathbb{L}} % 
\DeclareMathOperator{\re}{Re}         % real part
\DeclareMathOperator{\im}{Im}         % imaginary part
\DeclareMathOperator{\Ker}{Ker}
%
\DeclareMathOperator{\meet}{\wedge}
\DeclareMathOperator{\join}{\vee}
\DeclareMathOperator{\union}{\cup}
\DeclareMathOperator{\inter}{\cap}  % intersection
\DeclareMathOperator{\iso}{\cong}   % isomorphic
\DeclareMathOperator{\piso}{\simeq} % polynomially isomorphic
\newcommand{\Star}{^ *}
\DeclareMathOperator{\compose}{{\scriptstyle \circ}} % f o g
%
\newcommand{\wec}[1]{{\mathbf{#1}}}  % notation for vectors in algebras
%\newcommand{\wec}[1]{{\bar{#1}}} % which one do you prefer?
\newcommand{\wrel}[1]{\;#1\;}     % a\wrel{\alpha}b = a \alpha b
%              % this provides a flexible way to change the spacing
%
\newcommand{\m}[1]{{\mathbf{\uppercase{#1}}}}
\newcommand{\vr}[1]{{\mathcal{\uppercase {#1}}}}
%
\DeclareMathOperator{\Var}{\mathbf{V}}
\DeclareMathOperator{\Ho}{\mathbf{H}}
\DeclareMathOperator{\Su}{\mathbf{S}}
\DeclareMathOperator{\Pd}{\mathbf{P}}
\DeclareMathOperator{\Iso}{\mathbf{I}}
\DeclareMathOperator{\Ps}{\mathbf{P}_s}
\newcommand{\fin}[1]{{\mathbf{#1}}_{\text{\upshape fin}}}
%
\DeclareMathOperator{\Pol}{Pol}
\DeclareMathOperator{\Clo}{Clo}
\DeclareMathOperator{\Con}{Con}
\DeclareMathOperator{\Conb}{\mathbf{Con}} % Con(A) as a lattice
\DeclareMathOperator{\cg}{Cg}
\DeclareMathOperator{\sg}{Sg}
\newcommand{\Cg}[1]{{\cg}^{\m{#1}}}
\newcommand{\Sg}[1]{{\sg}^{\m{#1}}}
%
\newcommand{\M}[3]{\text{\upshape M}_{\m{#1}} (#2, #3)}
\newcommand{\minl}[1]{\text{\upshape Min}_{\m{#1}}}
%
\DeclareMathOperator{\typ}{typ}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\Sep}{Sep}
\DeclareMathOperator{\ssb}{ssb}
\DeclareMathOperator{\sst}{sst}
%
\DeclareMathOperator{\utyp}{\mathbf{1}}% unary      
\DeclareMathOperator{\atyp}{\mathbf{2}}% affine     
\DeclareMathOperator{\btyp}{\mathbf{3}}% boolean    
\DeclareMathOperator{\ltyp}{\mathbf{4}}% lattice    
\DeclareMathOperator{\styp}{\mathbf{5}}% semilattice
\DeclareMathOperator{\ityp}{\mathbf{i}}
\DeclareMathOperator{\jtyp}{\mathbf{j}}
%
\DeclareMathOperator{\solv}{\stackrel{s}{\sim}}
\DeclareMathOperator{\ssolv}{\stackrel{ss}{\sim}}
%
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}
%
\newcommand{\lb}{\langle} % <...>
\newcommand{\rb}{\rangle}
\newcommand{\lv}{{[\![}}  % [[...]]
\newcommand{\rv}{{]\!]}}
%
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
%
\newcommand{\vep}{\varepsilon}
\newcommand{\di}{\diamond}
%


\begin{document}

\noindent
{\large Transcendence Degree.}

\bigskip

\bigskip

\noindent
{\bf Definition.} If ${\BF} < {\BE}$ is a field extension, then
an element $\alpha\in {\BE}-{\BF}$ is {\bf {\em algebraic over 
$\BF$}} if $\alpha$ is a root of some nonzero $f(x)\in {\BF}[x]$. 
Otherwise $\alpha$ is {\bf {\em transcendental over 
$\BF$}}. A subset $X\subseteq {\BE}$ is 
{\bf {\em algebraically dependent over $\BF$}} 
if there is a nonzero polynomial 
$f(x_1,\ldots,x_n)\in {\BF}[x_1,\ldots,x_n]$,
and a sequence $(\alpha_1,\ldots,\alpha_n)$
of elements of $X$ for which $f(\alpha_1,\ldots,\alpha_n) = 0$.
Otherwise $X$ is {\bf {\em algebraically independent over $\BF$}}.

A {\bf {\em transcendence base}}  for ${\BE}$ over 
${\BF}$ is a maximal algebraically independent subset.
The {\bf {\em transcendence degree}} of an extension
is the size of a transcendence base.
The field $\BE$ is a 
{\bf {\em pure transcendental extension}} of $\BF$
if $\BE$ is generated over $\BF$ by a transcendence base.

\bigskip

\noindent
{\bf Examples.} 
\begin{itemize}
\item The elements $x, y, x+y$ and $xy$ are transcendental elements
of ${\BQ}(x,y)$.
\item The elements $x$ and $\sqrt{x}$ are algebraically dependent
transcendental elements 
of $\overline{{\BZ}_2(x)}$.
\item ${\BQ}(x,y)$ is a pure transcendental extansion of $\BQ$,
since it is generated over $\BQ$ by the set $\{x,y\}$ 
(which is a transcendence base for 
${\BQ}(x,y)$ over $\BQ$). The set $\{x,y,x+y\}$ is algebraically 
dependent, since the sequence $(x,y,x+y)$ satisfies the
equation $f = 0$ when $f$ is the nonzero 
polynomial $f(x_1,x_2,x_3) = x_1 + x_2 - x_3$.
\end{itemize}

\bigskip

\noindent
{\bf Facts.} 
\begin{itemize}
\item All transcendence bases have the same size.
\item The transcendence degree of an extension is zero
if and only if the extension is algebraic.
\item The transcendence degree of ${\BF}(x)$ over ${\BF}$
is $1$. The transcendence degree of $\BC$ over $\BQ$ is $2^{\aleph_0}$.
\item If ${\BF}<{\BK}<{\BE}$ are extensions, then the transcendence degree
of ${\BF}<{\BK}$  plus the transcendence degree of 
${\BK}<{\BE}$ equals the transcendence degree of 
${\BF}<{\BE}$.
\item For any extension ${\BF}<{\BE}$ there is an intermediate
field $\BK$ such that the extension 
${\BF}<{\BK}$ is purely transcendental and 
${\BK}<{\BE}$ is algebraic. (One can take $\BK$ to be any subfield 
of $\BE$ generated over $\BF$ by a transcendence base.)
\end{itemize}

\end{document}

