Let $\bar{\mathbf{Q}}$ be an algebraic closure of the rationals, and $\alpha$ denote an algebraic number in $\bar{\mathbf{Q}}$. We define the height of $\alpha$, denoted by $H(\alpha)$, to be $$H(\alpha) = \left( \prod_v \max(1,\Vert \alpha\Vert_v) \right)^{1/[K:\mathbf{Q}]}.$$ Here $K$ is a number field containing $\alpha$ and the product runs over the set of normalized valuations $v$ of $K$. For a non-empty finite subset $B\subset \bar{\mathbf{Q}}$, we define $$H(B) := \max \{H(\alpha) \ | \ \alpha\in B\}.$$

Now, let $K$ be a number field and let $\alpha$ be an algebraic number contained in $K$. Let $B$ be the set of conjugates of $\alpha$.

**Question.** Can we bound $H(B)$ from above in terms of data depending only on $\alpha$ and $K$?

**Example.** The number of elements of $B$ is less or equal to $[K:\mathbf{Q}]$.

I'm looking for a bound of the form $H(B)\leq H(\alpha)^{[K:\mathbf{Q}]}$ if possible.