Dan Ursu, Department of Pure Mathematics, University of Waterloo
"Ruler-compass constructions"
Start
with
two
distinct
points
on
the
plane.
Between
any
two
points,
you
can
draw
a
straight
line
passing
through
them,
or
a circle
centered
at one
and
passing
through the
other.
New
points
are
given
by
intersections
of
distinct
lines
and/or
circles. Repeat.
What
can
you
construct?
This
is
a
problem dating back
to
Euclid
and
the
gang.
They
were
able
to
bisect
an
arbitrary
angle,
but
not
trisect
one.
They were
able
to
construct
regular
polygons
with a
certain
number
of
sides,
but
not
others
(ex:
regular
pentagons
but
not
regular heptagons).
They
were
also
unable
to
construct
a
square
and
a
circle
having
the
same area.
These
problems
remained unanswered
for
two
millennia
until
the
19th
century.
In
this
talk,
we
will
view
ruler-compass
constructions
in
the
framework
of
field
theory,
and
see
how
answering
these
questions suddenly
becomes
quite
possible.
MC 5479