TITLE: Height of algebraic numbers
QUESTION [9 upvotes]: I would like to find effective upper bound for the height of $a+b$ and $a/b$ and $ab$ knowing the heights of $a$ and $b$. Thanks.

REPLY [2 votes]: I too was looking for the answer to the same question. It seems necessary to define "height" since there are of course several variants in use. The height $h(a)$ I am interested in is the maximum of the absolute values of the coefficients of the minimal polynomial of the algebraic number $a$. Note that the first answer refers to a different notion of height, since, for example,
$9 = h(9) = h(3 \cdot 3) \not \leq h(3) + h(3) = 3 + 3 = 6.$
I imagine there must be upper bounds of the form 
$h(ab) \leq f(d) h(a)^{g(d)} h(b)^{g(d)}$
for some simple functions $f$ and $g$, where $d$ is the degree of a field extension of $\mathbb{Q}$ containing both $a$ and $b$, for example.
Are there any such results in the literature, and similarly for $h(a+b)$ and $h(a/b)$?