Tuesday, May 28, 14:00, 7.527
Yichuan Yang (Beihang University, Beijing)
The complex number field can be directed partially ordered.
The absence of a compatible partial order on the complex number field
sets a full stop for many
important methods of real analysis. In fact,
by Artin-Schreier theory (1926) on formal real fields, it is known that
there does not exist a linear
order as in the case of the real numbers.
Birkhoff and Pierce (1956) asked whether there is a compatible lattice
order (which is much weaker than
a linear one) on the complex
field. Despite of considerable effort, this question is still open. In
many situations, however, the
existence of a directed order is enough to
discuss some important questions. For the complex field, I dealt with
that question in J. Alg., 2006.
The complete answer is given in J. Alg., 2011
(joint with N. Schwartz). More recently, for any field, we
can determine whether it can be directed
partially ordered or not.