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\newtheorem{thmd-rs}{Theorem~D.$\negthickspace$}  %% rs = Reinhard Schultz
\newtheorem{thm-rs}{Theorem}  %% rs = Reinhard Schultz
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\begin{document}
\setcounter{page}{25}

\begin{center}
{APPENDIX D}
\end{center}
\bigskip
\begin{center}
{\Large AUTOMORPHISMS OF THE COMPLEX NUMBERS}
\end{center}
\bigskip\bigskip

This appendix discusses an issue that arose in Section VI.??.
It is written at a level somewhat higher than the notes, but 
not as high as the level of material in Appendix B.  The necessary 
background can be found in the graduate algebra text by T. 
Hungerford and the introductory granduate level topology text by 
J. Munkres that are listed in the bibliography.
\bigskip

The Fundamental Theorem of Projective Geometry in Chapter
VI (Theorem VI.??) implies that for each positive integer
$n$ there is a group homomorphism from the group 
$\mathsf{Proj}_n(\bF)$ of geometric 
symmetries, or {\sl collineations\/}, of $\bF\P^n$ 
to the automorphism group of $\bF$, and this mapping
is onto.  For some fields such as the rational numbers
$\bQ$ and the real numbers $\bR$, the only automorphism
is the identity.  On the other hand, in the notes we
pointed out that the field $\bC$ of complex numbers also
has a second automorphism given by complex conjugation.
This leads immediately to the following:
\bigskip

{\bf Quesion. }
{\sl Are there any automorphisms of $\bC$ other than
the identity and complex conjutation?}
\bigskip

In fact, there are two ways of answering this question.
If we restrict our attention to automorphisms that
are continuous as mappings of $\bC\cong\bR^2$ to itself,
then the identity and complex conjugation are the only
possibilities.  On the other hand, if we consider 
arbitrary automorphisms of $\bC$ with no continuity
requirement, then the automorphism group of $\bC$
is huge.  One illustration of this is the formula
for its cardinality given below.
\bigskip\bigskip

\centerline{\it Continuous automorphisms of $\bC$}
\bigskip\bigskip

We shall first prove that there are only two 
continuous automorphisms.
\bigskip

\begin{thmd-rs}
Let $\varphi:\bC\to \bC$ be an automorphism of
the complex numbers that is also {\sl continuous}
as a function of two variables.  Then $\varphi$
is either equal to the identity or to complex
conjugation.
\end{thmd-rs}
\bigskip

{\bf Proof. }  Every automorphism sends 0 and 1
to themselves, and from this it follows that
every automorphism sends the rational numbers
$\bQ\subset \bC$ to itself by the identity.
\bigskip

Let $\bQ[\bfi]$ be the set of all complex numbers
of the form $a+b\bfi$, where $\bfi^2=-1$ and 
$a,~b\in \bQ$.  Since $\pm\,\bfi$ are the only
two complex numbers whose squares are equal to
$-1$, it follows that $\varphi(\bfi)~=~\pm\,\bfi$.
Therefore $\varphi$ agrees with either the identity
or conjugation on $\bQ[\bfi]$.
\bigskip

Just as the rationals are dense in $\bR$, so also
the set $\bQ[\bfi]$ is dense in $\bC$.  Since two 
continuous functions from $\bC$ to itself are equal
if they agree on a dense subset, it follows that
$\varphi$ is equal to the identity or complex 
conjugation, depending upon whether $\varphi
(\bfi)~=~\bfi$ or $\varphi[\bfi]~=~-\bfi$.$\blacksquare$
\bigskip\bigskip

\centerline{\it Arbitrary automorphisms of $\bC$}
\bigskip\bigskip

This discussion will rely heavily on the existence of 
{\sl transcendence bases} for the $\bC$ with respect
to its subfield $\bQ$.  Formal definitions and the
basic properties of such subsets are contained in 
Section VI.1 of Hungerford's book.  A brief summary
of some important points is given in the Wikipedia
reference:\footnote{Previous comments about Wikipedia
also apply here.}
\medskip

INSERT LINK HERE
\bigskip


%F/Q, cardinality of transcendence base = cardinality of F

%what if F=C?

%every 1--1 onto map of the transcendence base to itself
%extends to an automorphism of F if the latter is algebraically
%closed

%aut(C) contains elements other than identity and conjugation

%cardinality of aut(C)

TO BE COMPLETED
\bigskip

$\backepsilon$~~{\Large $\backepsilon$}~~$\pmb{\backepsilon}$~~
{\Large $\pmb{\backepsilon}$}
\medskip

$\backepsilon$~~$\obr$~~$\obrt$
\medskip

$i\boldsymbol{\backepsilon}$~~$\boldsymbol{\obr}$~~$\boldsymbol{\obrt}$
\medskip

${\text{\tencyr B}}$~
${\text{\tencyr zh}}$~
${\text{\tencyr ch}}$~
${\text{\tencyr yu}}$~
${\text{\tencyr ya}}$~
\medskip

${\text{\tencyi B}}$~
${\text{\tencyi d}}$~
${\text{\tencyi dj}}$~
${\text{\tencyi Dj}}$~
${\text{\tencyi zh}}$~
${\text{\tencyi ch}}$~
${\text{\tencyi yu}}$~
${\text{\tencyi ya}}$~
\medskip

${\text{\elfcyr B}}$~
${\text{\elfcyr zh}}$~
${\text{\elfcyr ch}}$~
${\text{\elfcyr yu}}$~
${\text{\elfcyr ya}}$~
\medskip
\medskip

${\text{\elfcyi B}}$~
${\text{\elfcyi b}}$~
${\text{\elfcyi d}}$~
${\text{\elfcyi dj}}$~
${\text{\elfcyi Dj}}$~
${\text{\elfcyi zh}}$~
${\text{\elfcyi ch}}$~
${\text{\elfcyi yu}}$~
${\text{\elfcyi ya}}$~
\medskip

$\pmb{\text{\elfcyi B}}$~
$\pmb{\text{\elfcyi b}}$~
$\pmb{\text{\elfcyi v}}$~
$\pmb{\text{\elfcyi d}}$~
$\pmb{\text{\elfcyi dj}}$~
$\pmb{\text{\elfcyi Dj}}$~
$\pmb{\text{\elfcyi zh}}$~
$\pmb{\text{\elfcyi Zh}}$~
$\pmb{\text{\elfcyi nj}}$~
$\pmb{\text{\elfcyi Nj}}$~
$\pmb{\text{\elfcyi ch}}$~
$\pmb{\text{\elfcyi Ch}}$~
$\pmb{\text{\elfcyi yu}}$~
$\pmb{\text{\elfcyi ya}}$~
$\pmb{\text{\elfcyi Yu}}$~
$\pmb{\text{\elfcyi Ya}}$~
$\pmb{\text{\elfcyi ts}}$~
$\pmb{\text{\elfcyi Ts}}$~
$\pmb{\text{\elfcyi sh}}$~
$\pmb{\text{\elfcyi Sh}}$~
$\pmb{\text{\elfcyi z}}$~
$\pmb{\text{\elfcyi Z}}$~
\medskip

${\text{\elfcyi B}}$~
${\text{\elfcyi b}}$~
${\text{\elfcyi v}}$~
${\text{\elfcyi p}}$~
${\text{\elfcyi zh}}$~
${\text{\elfcyi ch}}$~
${\text{\elfcyi yu}}$~
${\text{\elfcyi ya}}$~
${\text{\elfcyi \`e}}$~
${\text{\elfcyi \`E}}$~
\medskip

$\pmb{\text{\twcyi B}}$~
$\pmb{\text{\twcyi b}}$~
$\pmb{\text{\twcyi v}}$~
$\pmb{\text{\twcyi d}}$~
$\pmb{\text{\twcyi dj}}$~
$\pmb{\text{\twcyi Dj}}$~
$\pmb{\text{\twcyi p}}$~
$\pmb{\text{\twcyi zh}}$~
$\pmb{\text{\twcyi Zh}}$~
$\pmb{\text{\twcyi nj}}$~
$\pmb{\text{\twcyi Nj}}$~
$\pmb{\text{\twcyi ch}}$~
$\pmb{\text{\twcyi Ch}}$~
$\pmb{\text{\twcyi yu}}$~
$\pmb{\text{\twcyi ya}}$~
$\pmb{\text{\twcyi Yu}}$~
$\pmb{\text{\twcyi Ya}}$~
$\pmb{\text{\twcyi ts}}$~
$\pmb{\text{\twcyi Ts}}$~
$\pmb{\text{\twcyi sh}}$~
$\pmb{\text{\twcyi Sh}}$~
$\pmb{\text{\twcyi z}}$~
$\pmb{\text{\twcyi Z}}$~
\medskip

${\text{\twcyi B}}$~
${\text{\twcyi b}}$~
${\text{\twcyi v}}$~
${\text{\twcyi p}}$~
${\text{\twcyi zh}}$~
${\text{\twcyi Zh}}$~
${\text{\twcyi nj}}$~
${\text{\twcyi Nj}}$~
${\text{\twcyi ch}}$~
${\text{\twcyi Ch}}$~
${\text{\twcyi yu}}$~
${\text{\twcyi ya}}$~
${\text{\twcyi Yu}}$~
${\text{\twcyi Ya}}$~
${\text{\twcyi ts}}$~
${\text{\twcyi Ts}}$~
${\text{\twcyi sh}}$~
${\text{\twcyi Sh}}$~
${\text{\twcyi z}}$~
${\text{\twcyi Z}}$~
\medskip

$\pmb{\text{\twcyss B}}$~
$\pmb{\text{\twcyss b}}$~
$\pmb{\text{\twcyss v}}$~
$\pmb{\text{\twcyss d}}$~
$\pmb{\text{\twcyss dj}}$~
$\pmb{\text{\twcyss Dj}}$~
$\pmb{\text{\twcyss p}}$~
$\pmb{\text{\twcyss zh}}$~
$\pmb{\text{\twcyss Zh}}$~
$\pmb{\text{\twcyss nj}}$~
$\pmb{\text{\twcyss Nj}}$~
$\pmb{\text{\twcyss ch}}$~
$\pmb{\text{\twcyss Ch}}$~
$\pmb{\text{\twcyss yu}}$~
$\pmb{\text{\twcyss ya}}$~
$\pmb{\text{\twcyss Yu}}$~
$\pmb{\text{\twcyss Ya}}$~
$\pmb{\text{\twcyss ts}}$~
$\pmb{\text{\twcyss Ts}}$~
$\pmb{\text{\twcyss sh}}$~
$\pmb{\text{\twcyss Sh}}$~
$\pmb{\text{\twcyss z}}$~
$\pmb{\text{\twcyss Z}}$~
\medskip

${\text{\twcyss B}}$~
${\text{\twcyss b}}$~
${\text{\twcyss v}}$~
${\text{\twcyss p}}$~
${\text{\twcyss zh}}$~
${\text{\twcyss Zh}}$~
${\text{\twcyss nj}}$~
${\text{\twcyss Nj}}$~
${\text{\twcyss ch}}$~
${\text{\twcyss Ch}}$~
${\text{\twcyss yu}}$~
${\text{\twcyss ya}}$~
${\text{\twcyss Yu}}$~
${\text{\twcyss Ya}}$~
${\text{\twcyss ts}}$~
${\text{\twcyss Ts}}$~
${\text{\twcyss sh}}$~
${\text{\twcyss Sh}}$~
${\text{\twcyss z}}$~
${\text{\twcyss Z}}$~
\medskip

$\pmb{\text{\twcysc B}}$~
$\pmb{\text{\twcysc b}}$~
$\pmb{\text{\twcysc v}}$~
$\pmb{\text{\twcysc d}}$~
$\pmb{\text{\twcysc dj}}$~
$\pmb{\text{\twcysc Dj}}$~
$\pmb{\text{\twcysc p}}$~
$\pmb{\text{\twcysc zh}}$~
$\pmb{\text{\twcysc Zh}}$~
$\pmb{\text{\twcysc nj}}$~
$\pmb{\text{\twcysc Nj}}$~
$\pmb{\text{\twcysc ch}}$~
$\pmb{\text{\twcysc Ch}}$~
$\pmb{\text{\twcysc yu}}$~
$\pmb{\text{\twcysc ya}}$~
$\pmb{\text{\twcysc Yu}}$~
$\pmb{\text{\twcysc Ya}}$~
$\pmb{\text{\twcysc ts}}$~
$\pmb{\text{\twcysc Ts}}$~
$\pmb{\text{\twcysc sh}}$~
$\pmb{\text{\twcysc Sh}}$~
$\pmb{\text{\twcysc z}}$~
$\pmb{\text{\twcysc Z}}$~
\medskip

${\text{\twcysc B}}$~
${\text{\twcysc b}}$~
${\text{\twcysc v}}$~
${\text{\twcysc p}}$~
${\text{\twcysc zh}}$~
${\text{\twcysc Zh}}$~
${\text{\twcysc nj}}$~
${\text{\twcysc Nj}}$~
${\text{\twcysc ch}}$~
${\text{\twcysc Ch}}$~
${\text{\twcysc yu}}$~
${\text{\twcysc ya}}$~
${\text{\twcysc Yu}}$~
${\text{\twcysc Ya}}$~
${\text{\twcysc ts}}$~
${\text{\twcysc Ts}}$~
${\text{\twcysc sh}}$~
${\text{\twcysc Sh}}$~
${\text{\twcysc z}}$~
${\text{\twcysc Z}}$~
\medskip

$\pmb{\text{\catcyi B}}$~
$\pmb{\text{\catcyi b}}$~
$\pmb{\text{\catcyi v}}$~
$\pmb{\text{\catcyi d}}$~
$\pmb{\text{\catcyi dj}}$~
$\pmb{\text{\catcyi Dj}}$~
$\pmb{\text{\catcyi p}}$~
$\pmb{\text{\catcyi zh}}$~
$\pmb{\text{\catcyi Zh}}$~
$\pmb{\text{\catcyi nj}}$~
$\pmb{\text{\catcyi Nj}}$~
$\pmb{\text{\catcyi ch}}$~
$\pmb{\text{\catcyi Ch}}$~
$\pmb{\text{\catcyi yu}}$~
$\pmb{\text{\catcyi ya}}$~
$\pmb{\text{\catcyi Yu}}$~
$\pmb{\text{\catcyi Ya}}$~
$\pmb{\text{\catcyi ts}}$~
$\pmb{\text{\catcyi Ts}}$~
$\pmb{\text{\catcyi sh}}$~
$\pmb{\text{\catcyi Sh}}$~
$\pmb{\text{\catcyi z}}$~
$\pmb{\text{\catcyi Z}}$~
\medskip

${\text{\catcyi B}}$~
${\text{\catcyi b}}$~
${\text{\catcyi v}}$~
${\text{\catcyi p}}$~
${\text{\catcyi zh}}$~
${\text{\catcyi Zh}}$~
${\text{\catcyi nj}}$~
${\text{\catcyi Nj}}$~
${\text{\catcyi ch}}$~
${\text{\catcyi Ch}}$~
${\text{\catcyi yu}}$~
${\text{\catcyi ya}}$~
${\text{\catcyi Yu}}$~
${\text{\catcyi Ya}}$~
${\text{\catcyi ts}}$~
${\text{\catcyi Ts}}$~
${\text{\catcyi sh}}$~
${\text{\catcyi Sh}}$~
${\text{\catcyi z}}$~
${\text{\catcyi Z}}$~
\medskip

$\pmb{\text{\catcyss B}}$~
$\pmb{\text{\catcyss b}}$~
$\pmb{\text{\catcyss v}}$~
$\pmb{\text{\catcyss d}}$~
$\pmb{\text{\catcyss dj}}$~
$\pmb{\text{\catcyss Dj}}$~
$\pmb{\text{\catcyss p}}$~
$\pmb{\text{\catcyss zh}}$~
$\pmb{\text{\catcyss Zh}}$~
$\pmb{\text{\catcyss nj}}$~
$\pmb{\text{\catcyss Nj}}$~
$\pmb{\text{\catcyss ch}}$~
$\pmb{\text{\catcyss Ch}}$~
$\pmb{\text{\catcyss yu}}$~
$\pmb{\text{\catcyss ya}}$~
$\pmb{\text{\catcyss Yu}}$~
$\pmb{\text{\catcyss Ya}}$~
$\pmb{\text{\catcyss ts}}$~
$\pmb{\text{\catcyss Ts}}$~
$\pmb{\text{\catcyss sh}}$~
$\pmb{\text{\catcyss Sh}}$~
$\pmb{\text{\catcyss z}}$~
$\pmb{\text{\catcyss Z}}$~
\medskip

${\text{\catcyss B}}$~
${\text{\catcyss b}}$~
${\text{\catcyss v}}$~
${\text{\catcyss p}}$~
${\text{\catcyss zh}}$~
${\text{\catcyss Zh}}$~
${\text{\catcyss nj}}$~
${\text{\catcyss Nj}}$~
${\text{\catcyss ch}}$~
${\text{\catcyss Ch}}$~
${\text{\catcyss yu}}$~
${\text{\catcyss ya}}$~
${\text{\catcyss Yu}}$~
${\text{\catcyss Ya}}$~
${\text{\catcyss ts}}$~
${\text{\catcyss Ts}}$~
${\text{\catcyss sh}}$~
${\text{\catcyss Sh}}$~
${\text{\catcyss z}}$~
${\text{\catcyss Z}}$~
\medskip

$\pmb{\text{\catcysc B}}$~
$\pmb{\text{\catcysc b}}$~
$\pmb{\text{\catcysc v}}$~
$\pmb{\text{\catcysc d}}$~
$\pmb{\text{\catcysc dj}}$~
$\pmb{\text{\catcysc Dj}}$~
$\pmb{\text{\catcysc p}}$~
$\pmb{\text{\catcysc zh}}$~
$\pmb{\text{\catcysc Zh}}$~
$\pmb{\text{\catcysc nj}}$~
$\pmb{\text{\catcysc Nj}}$~
$\pmb{\text{\catcysc ch}}$~
$\pmb{\text{\catcysc Ch}}$~
$\pmb{\text{\catcysc yu}}$~
$\pmb{\text{\catcysc ya}}$~
$\pmb{\text{\catcysc Yu}}$~
$\pmb{\text{\catcysc Ya}}$~
$\pmb{\text{\catcysc ts}}$~
$\pmb{\text{\catcysc Ts}}$~
$\pmb{\text{\catcysc sh}}$~
$\pmb{\text{\catcysc Sh}}$~
$\pmb{\text{\catcysc z}}$~
$\pmb{\text{\catcysc Z}}$~
\medskip

${\text{\catcysc B}}$~
${\text{\catcysc b}}$~
${\text{\catcysc v}}$~
${\text{\catcysc p}}$~
${\text{\catcysc zh}}$~
${\text{\catcysc Zh}}$~
${\text{\catcysc nj}}$~
${\text{\catcysc Nj}}$~
${\text{\catcysc ch}}$~
${\text{\catcysc Ch}}$~
${\text{\catcysc yu}}$~
${\text{\catcysc ya}}$~
${\text{\catcysc Yu}}$~
${\text{\catcysc Ya}}$~
${\text{\catcysc ts}}$~
${\text{\catcysc Ts}}$~
${\text{\catcysc sh}}$~
${\text{\catcysc Sh}}$~
${\text{\catcysc z}}$~
${\text{\catcysc Z}}$~
\medskip

$B\hoch A$\qquad$B\tothe A$

\end{document}
If $(S,\Pi,d)$ is a geometrical incidence $n$-space, then 
the methods and results of Section IV.4 show that there
is a dual geometrical incidence $n$-space 
$(S,\Pi,d)$ is a geometrical incidence whose points
are the hyperplanes of $S$, and such that there is 
a canonical 1--1 correspondence between the $k$-planes
of $S^*$ and the $(n-k-1)$-planes of $S$ for all
$k$ such that $-1\leq k\leq n$.  If $\bF$ is a
(commutative) field and $S = \bF\bP^n$, then the 
material on analytic projective geometry in this notes
shows that $S^*$ is also isomorphic to $S = \bF\bP^n$.
It is natural to ask if $(S,\Pi,d)$ and its projective dual 
$(S^*,\Pi^*,d^*)$ are {\bf always} isomorphic to each other.  
The purpose of this Appendix is to discuss this problem
and the related algebraic questions.  Unfortunately, the 
discussion ultimately involves algebraic input that goes 
far beyond the undergraduate level, so we shall merely sketch 
the main ideas and give references for additional information
on the underlying algebra.
\bigskip
\end{document}
If $(S,\Pi,d)$ is a geometrical incidence $n$-space, then 
the methods and results of Section IV.4 show that there
is a dual geometrical incidence $n$-space 
$(S,\Pi,d)$ is a geometrical incidence whose points
are the hyperplanes of $S$, and such that there is 
a canonical 1--1 correspondence between the $k$-planes
of $S^*$ and the $(n-k-1)$-planes of $S$ for all
$k$ such that $-1\leq k\leq n$.  If $\bF$ is a
(commutative) field and $S = \bF\bP^n$, then the 
material on analytic projective geometry in this notes
shows that $S^*$ is also isomorphic to $S = \bF\bP^n$.
It is natural to ask if $(S,\Pi,d)$ and its projective dual 
$(S^*,\Pi^*,d^*)$ are {\bf always} isomorphic to each other.  
The purpose of this Appendix is to discuss this problem
and the related algebraic questions.  Unfortunately, the 
discussion ultimately involves algebraic input that goes 
far beyond the undergraduate level, so we shall merely sketch 
the main ideas and give references for additional information
on the underlying algebra.
\bigskip

We begin by summarizing a point mentioned in Example 
2 from Section IV.4 (and discussed further in Sections
V.1 and VI.1).
If $\bF$ is a skew-field and $n\geq 2$ is an integer, then
the dual projective $n$-space $\left(\bF\bP^n\right)^*$ 
associated to
$\bF\bP^n$ is isomorphic to $(\Fop)\bP^n$, where $\Fop$
is the skew-field obtained from $\bF$ by reversing the
order of multiplication in $\bF$.  Of course, if $\bF$ is
commutative, then the multiplications in $\bF$ and $\Fop$
are identical, and therefore it follows that $S^*$ is
isomorphic to $S$ in such cases.  In order to handle the
general situation, we must describe the relationship between 
the two skew-fields $\bF$ and $\Fop$ in the noncommutative 
case and derive its implications for the relationship between 
$\bF\bP^n$ and the isomorphic dual objects
$(\Fop)\bP^n \cong \left(\bF\bP^n\right)^*$.
\bigskip

In Appendix A we described a skew-field $\bK$ called the
{\sl quaterions} which is not commutative, and thus it is
natural to ask whether $\bK\bP^n$ and $(\Kop)\bP^n \cong 
\left(\bK\bP^n\right)^*$ are isomorphic.  By the general 
classification results, this will happen if and only if 
there is an algebraic isomorphism between $\bK$ and $\Kop$;
note that since $\bK$ is not commutative 
this map cannot be the identity.
\bigskip

In fact, there is an operation on $\bK$
called {\sl quaternionic conjugation} which is defined by 
$$(a_1\;+\;a_2\bfi\;+\;a_3\bfj\;+\;a_4\bfk)^*~~=~~
a_1\;-\;a_2\bfi\;-\;a_3\bfj\;-\;a_4\bfk$$
(see Birkhoff and MacLane, pp. 222--225, especially 
Exercise 5, p. 224) which has the following properties:
$$\bigl(\,\bfx^*\,\bigr)^*~~=~~\bfx\qquad\quad
(\bfx\;+\;\bfy)^*~~=~~\bfx^*~+~\bfy^*\qquad\quad
(\bfx\,\cdot\,\bfy)^*~~=~~\bfy^*\,\cdot\,\bfx^*$$
Note that conjugation, like matrix transposition, reverses
the order of multiplication.  This  conjugation map
is 1--1 and onto by the first identity in the display,
and the  other two identities imply it defines an
isomorphism from $\bK$ to $\Kop$.
\bigskip

More generally, by Theorem V.10 we have the following:
\bigskip

\begin{thm-rs}
Let $\bF$ be a skew-field and let $S=\bF\bP^n$, where 
$n\geq 2$.  Then $S$ is isomorphic to its projective dual space
$S^*$ if and only if $\bF$ is isomorphic to $\Fop$.
\end{thm-rs}
\bigskip

Therefore {\sl the existence of projective incidence $n$-spaces
that are not isomorphic to their projective duals} will follow 
{\bf if}  {\sl there
are skew-fields $\bF$ such that $\bF$ is not isomorphic to
$\Fop$\/}.
\bigskip
 
Although there are many examples of skew-fields $\bF$ for which 
$\bF$  and $\Fop$ are not isomorphic, describing them is quite 
nontrivial.
The simplest example of this type is given on pages 31--38 of
the book by Blanchard listed below; in fact, the exercises on
pages 37--38 yield a direct proof that $\bF$ and $\Fop$ are
not isomorphic in this case.  However, we shall also outline
a more comprehensive approach which yields this fact for Blanchard's
example and  allows one to find systematic infinite families  of
such skew-fields.
\bigskip

Given a skew-field $\bL$, its {\sl center} consists of all 
$a\in \bL$ such that $a\cdot x~=~x\cdot a$ for all $x\in\bL$.
This center is always nontrivial, and in particular it
contains the smallest subfield containing the unit element
(which is isomorphic to either the rational numbers $\bQ$
or one of the finite fields $\bZ_p$).
It is immediate that the center is a commutative subfield 
that we shall call $\bF$, and it also follows that $\bL$ is
a vector space over $\bF$.  For example, if $\bL$ is the
quaternions, then $\bF$ is the real numbers.  Conversely, 
given a field $\bF$ we can consider all skew-fields $\bL$
with center $\bF$ such that $\bL$ is a {\sl finite-dimensional}
vector space over $\bF$.  Since each such $\bL$ is isomorphic
to a subring of a finite-dimensional matrix algebra over  
$\bF$ (see Birkhoff and MacLane, pp. 226--227), the isomorphism
classes of such objects form a set.  A basic construction in
algebra makes this set into an abelian group called the {\bf
Brauer group} of $\bF$ and written $\mathrm{Br}(\bF)$.  The
details of the definition appear in the books by Adamson, 
Herstein, and Jacobson, and also in the papers by Serre (all
of which are listed below), and the book by Gille and Szamuely
covers the subject thoroughly (but at an extremely high level).
The following online references
might also be helpful:\footnote{In any citation of Wikepedia 
articles, it is important to recognize concerns about the 
accuracy of articles that are submitted by volunteers and 
subject to editing by a vast number of individuals whose views 
or understanding may be controversial or unreliable.  This 
issue has been noted explicitly by Wikipedia in its article on 
itself, where the issue is discussed in some detail.  However, for 
our purposes such questions do not cause problems because I have 
seen the articles cited below and found them to be accurate.}
\bigskip

\centerline{\tt http://en.wikipedia.org/wiki/Division{\_\_}ring}
\medskip

\centerline{\tt http://en.wikipedia.org/wiki/Field{\_\_}theory{\_\_}(mathematics)}
\medskip

\centerline{\tt http://en.wikipedia.org/wiki/Quaternion}
\medskip

\centerline{\tt http://en.wikipedia.org/wiki/Brauer{\_\_}group}
\medskip

\begin{quotation}
{\small (Note that there are {\bf two} underscore characters
separating words in these links.)}
\end{quotation}
\bigskip

The construction of $\mathrm{Br}(\bF)$ has the following
immediate consequence:
\bigskip

\begin{thm-rs}
Suppose that the commutative field $\bF$ admits no automorphisms
other than the identity (for example, this holds if $\bF$ is 
the rational numbers $\bQ$ or the real numbers $\bR$).  Let
$\bL$ be a skew-field $\bL$ which is finite-dimensional over
its center $\bF$, and assume further that $\bL$ is isomorphic
to $Lop$. Then $\bL$ determines an element of order $2$ in
the Brauer group $\mathrm{Br}(\bF)$.$\blacksquare$
\end{thm-rs}
\bigskip

Thus the proof that $\bL$ is not isomorphic to $\Lop$ for
Blanchard's example reduces to showing the following:
\medskip

\begin{enumerate}
\item[$(i)$]
The center of $\bL$ is the rational numbers $\bQ$.
\medskip

\item[$(ii)$]
The order of $\bL$ in the Brauer group is greater than 2.
\end{enumerate}
\medskip

The first assertion is verified in Blanchard's book, and the
second one follows from two easily stated facts:
\medskip

\begin{enumerate}
\item[$(iii)$]
By construction, the dimension of $\bL$ over $\bQ$ is equal
to 9.
\medskip

\item[$(iv)$]
By the results on pages 93--95 of Artin, Nesbitt and Thrall,
the order $m$ of $\bL$ in $\mathrm{Br}(\bF)$ satisfies
$m^2=\dim_\bQ\,\bL$, so that $m~=~3$.  In particular, 
$\bL$ determines an element of odd order in the Brauer
group of $\bQ$ and hence cannot be isomorphic to
$\Lop$.
\end{enumerate}
\bigskip

In fact, very powerful methods exist for calculating Brauer
groups, and these objects turn out to be fundamentally important
to an algebraic subject called {\sl class field theory\/}.
Numerous references are given below.  One consequence is that
elements of order $p$ exist in $\mathrm{Br}(\bQ)$ for every
prime $p$ (compare Adamson, pp. 220--221).
\bigskip\bigskip

\centerline{\sf REFERENCES FOR APPENDIX B}
\bigskip\bigskip

{\bf I. T. Adamson. }  {\sl Rings, Modules, and Algebras\/}.
Oliver and Boyd, Edinburgh, 1971.
\bigskip

{\bf E. Artin, C. Nesbitt, and R. Thrall. }  {\sl Rings with
Minimum Condition\/}.  University of Michigan Press, Ann 
Arbor, 1944.
\bigskip

{\bf A. Blanchard. } {\sl Les corps noncommutatifs\/}.
SUP Section ``Le math\'ematicien'' No. 9.  Presses Universitaires
de France, Paris, 1977.
\bigskip

{\bf P. Gille and T. Szamuely. } {\sl Central Simple Algebras 
and Galois Cohomology\/}. Cambridge Studies in Advanced. 
Mathematics No. 101.  Cambridge University Press, Cambridge, U.K., 
and New York, 2006.
\bigskip

{\bf I. N. Herstein. }  {\sl Non-commutative Rings\/}.
Carus Mathematical Monographs No. 15.  Mathematical Association
of America, Washington, DC, 1968.
\bigskip

{\bf N. Jacobson. } {\sl Structure of Rings\/}.  American
Mathematical Society Colloquium Publications No. 37.  American
Mathematical Society, Providence, RI, 1964.
\bigskip

{\bf J.-P. Serre. } {\it Applications alg\'ebriques de la
cohomologie des groupes {\rm II -- III}}.  Th\'eorie des alg\`ebres
simples, S\'eminaire H. Cartan {\bf 3} (1950/51),
Expos\'es 6 -- 7.  S\'ecretariat math\'ematique,
Paris, 1955. ---  {\sl Available online at the following 
site}:\footnote{In this link, there are {\bf two} underscore 
characters before the 3.}

\smallskip

{\tt http://www.numdam.org/numdam-bin/feuilleter?id=3DSHC{\_}1950-1951{\_}{\_}3{\_}}
\bigskip

{\bf J.-P. Serre. } Local Class Field Theory.  {\sl Algebraic 
Number Theory} (Proceedings of the NATO Advanced Study 
Institute, 1965), pp. 129--163.  Thompson Publishing, 
Washington, DC, 1967.
 



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