Q:

What is the lowest degree of a polynomial with integer coefficients which has a specific root.

What is the lowest possible degree of a polynomial $p(x)$ with integer coefficients which has one root 
$$x = \sum_{n=1}^k\sqrt{a_n}$$ where $1 \lt a_1 \lt \cdots \lt a_k$ are non-square integers?

A:

The minimal degree can be $2^k$, e.g., for $a_n$ being distinct prime numbers. We see that
$\sqrt{p_1}+\sqrt{p_2}+\cdots +\sqrt{p_k}$ has degree $2^k$ over $\mathbb{Q}$, and we have
$$
[\mathbb{Q}(\sqrt{p_1},\ldots ,\sqrt{p_k}):\mathbb{Q}]=[\mathbb{Q}(\sqrt{p_1}+\sqrt{p_2}+\cdots +\sqrt{p_k}):\mathbb{Q}]=2^k.
$$
This is explained here for a case with $k=2$.

