Minimal Mahler measure in number fields (Kate Petersen, University of Minnesota Duluth)

The Mahler measure of a polynomial is the modulus of its leading term multiplied by the moduli of all roots outside the unit circle.  The Mahler measure of an algebraic number b, M(b) is the Mahler measure of its minimal polynomial. By a result of Kronecker, an algebraic number b satisfies M(b)=1 if and only if b is a root of unity. Famously, Lehmer asked if there are algebraic numbers with Mahler measures arbitrarily close to 1 (but not equal to 1). We will investigate the minimal Mahler measure of a number field.  For a number field K this is the smallest Mahler measure of a non-torsion generator for K, written M(K). There are known upper and lower bounds for M(K) in terms of the degree and discriminant of K.  Focusing on cubics, we will discuss how these bounds correspond to other properties of the number field, and the sharpness of these bounds.  This is joint work with Lydia Eldredge.