Milner 317

Wednesdays, 12:30-1:30 PM

University of Texas at Austin

Abstract in PDF

**Wednesday, April 18
Milner 317, 12:30 pm**

**Title: **The
Weil height of the centralizer of complex conjugation

**Abstract:** We develope
the optimal abstraction of a well known inequality due A. Schinzel
concerning the Weil height of totally real algebraic numbers. Our new
result allows for consideration of arbitrary algebraic numbers having a
real Galois conjugate. As a consequence, we can associate a lower
bound for the Weil height of an algebraic number based on the index of
the centralizer of complex conjugation in the Galois group of the
Galois closure of the number. This interpretation is an
abstraction of a result due to F. Amoroso and R. Dvornicich. We will
discuss recent and related results due to other authors and new
research questions that have come to our attention.

Time permitting we will discuss some
recent progress of ours concerning Lehmer's problem. In
particular we will review our discovery that amongst all polynomials in
$\Z[x]$ whose splitting fields are contained in dihedral Galois
extensions of $\Q$, the lowest Mahler measure (other than 1) is
uniquely obtained by $\pm (x^3-x-1)$. This new result raises
promising research questions which we hope to discuss.