Timeline for Height of algebraic numbers

4 events

when toggle format	what		by	license	comment
Dec 17, 2016 at 17:53	comment	added	Joe Silverman		@KevinO'Bryant That's only correct if $\alpha$ is an algebraic integer. For algebraic numbers, one either needs to also use $p$-adic embeddings, or take the norm of the "denominator ideal" of $\alpha$. To see why, consider $\alpha=\frac12$. There are no complex embeddings for which $\alpha$ lies outside the unit circle, but $H_{\mathbb Q}(\alpha)=2$, not $1$.
May 29, 2012 at 13:55	comment	added	Kevin O'Bryant		$h(a)$ is also called the absolute logarithmic height of $a$, defined by $1/[K:Q] \log H_K(a)$, where $H_K(a)$ is the product of all of the embeddings (with multiplicity) of $a$ that land outside the complex unit circle. In other words, $h(a) = 1/d M(a)$, where $M(a)$ is the Mahler measure of $a$.
May 11, 2011 at 18:35	history	edited	Oleg Eroshkin	CC BY-SA 3.0	deleted 2 characters in body
May 11, 2011 at 18:14	history	answered	Oleg Eroshkin	CC BY-SA 3.0