Amador Martin-Pizarro, University of Freiburg
"Simplicity of the automorphism group of fields with operators"
In
1992
Lascar
proved
that
the
group
of
field
automorphisms
of
the
complex
numbers
which
fix
pointwise
the
algebraic
closure
of
the
rationals
is
simple,
assuming
the
continuum
hypothesis.
His
proof
used
strongly
the
topological
features
of
the
group
of
automorphisms
of
a
countable
structure,
as
a
Polish
group.
In
1997
Lascar
gave
a
different
proof
of
the
above,
without
assuming
the
continuum
hypothesis.
The
new
proof
needed
just
the
stability
of
the
theory
of
the
field
of
complex
numbers
(and
particularly
stationarity
of
types
as
a
way
to
merge
two
elementary
maps)
as
well
as
the
fact
that
the
field
of
complex
numbers
is
saturated
in
its
own
cardinality.
In
a
recent
preprint
with
T.
Blossier,
Z.
Chatzidakis
and
C.
Hardouin,
we
have
adapted
a
proof
of
Lascar
to
show
that
certain
groups
of
automorphisms
of
various
theories
of
fields
with
operators
are
simple.
It
particularly
applies
to
the
theory
of
difference
closed
fields,
which
is
simple
and
hence
has
possibly
no
models
which
are
saturated
in
their
uncountable
cardinality.
MC 5479