Absolute multiplicative Weil height
= Absolute multiplicative Weil height
{c}
{title2=$H(\alpha)$}
{wiki=Height_function}
For an algebraic number $\alpha$ in a number field $L$, the absolute multiplicative Weil height is
$$
H(\alpha)=\prod_{v\in M_L}\max(1,|\alpha|_v)^{[L_v:\mathbb Q_v]/[L:\mathbb Q]}.
$$
It is independent of the field $L$ containing $\alpha$, satisfies $H(\alpha^n)=H(\alpha)^{|n|}$, and obeys $H(\alpha+\beta)\leq2H(\alpha)H(\beta)$.