Let $L/\mathbb{Q} $ be a Galois extension of degree $n$ let $\mathbb{Q}(\alpha)$ be an intermidiate field extension of $L/\mathbb{Q} $ where $\alpha$ is an algebraic numbers with minimal polynomial $P$.let $\mathcal{N}_{L/\mathbb{Q}}(\alpha)$ be the norm of $\alpha $ and $H(P)$ is the absolute maximum of the coefficiencts of $P$ .
I am looking for an upper bound for $|\mathcal{N}_{L/\mathbb{Q}}(\alpha)|$ In terms of $H(P)$ and $n$ .
The norm $\mathcal N_{L/\mathbb Q}(\alpha)$ is the product over all $\sigma \in \mathrm{Gal}(L/\mathbb Q)$ of $\sigma(\alpha)$. If $m$ is the degree of $\alpha$ over $\mathbb Q$, then each of the $m$ algebraic conjugates of $\alpha$ appears exactly $n/m$ times in this product.
If $P(x) = x^m + c_{m-1} x^{m-1} + \cdots + c_1 x + c_0$, then the product of all the conjugates of $\alpha$ is $(-1)^m c_0$. It follows that we have an exact formula for the norm in terms of $P$: $$ \mathcal N_{L/\mathbb Q}(\alpha) = \left( (-1)^m c_0 \right)^{n/m} = (-1)^n c_0^{n/m}. $$ This is bounded in absolute value by $H(P)^{n/m}$, or by $H(P)^n$ if you really want to avoid dependence on $m$.