# On Heights of Algebraic Numbers

## Document Type

Lecture

## Department

Mathematics (CMC)

## Publication Date

3-3-2009

## Abstract

Weil height h of an algebraic number z measures its "arithmetic complexity", and h(z) is always non-negative. In fact, h(z) = 0 if and only if z is a root of unity. So suppose z is an algebraic number of degree d which is not a root of unity. How small can h(z) be? A famous conjecture of D. H. Lehmer (1932) states that h(z) cannot be arbitrarily close to 0, in fact there is (conjecturally) a gap between 0 and the smallest height value of an algebraic number of degree d, where this gap depends on d. There are many results in the direction Lehmer's conjecture, although the conjecture is still open. We will discuss Lehmer's conjecture, some related results, and a fascinating development of Zhang, Zagier, and others (mid-90's) on height restrictions for points on certain curves.

## Rights Information

© 2009 Lenny Fukshansky

## Terms of Use & License Information

## Recommended Citation

Fukshansky, Lenny. "On Heights of Algebraic Numbers." Algebra/Number Theory/Combinatorics Seminar, Claremont Colleges, Claremont, California. 3 March 2009.

## Comments

This lecture was given during the Algebra/Number Theory/Combinatorics Seminar at the Claremont Colleges in March 2009.