# 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?

• The answer will depend on common factors for the $a_i$. For example, if $a_1=2$ and $a_2=8$, you have a linear factor with the root you want. But that will not work for general $a_1$, $a_2$. – Paul Mar 3 '15 at 21:46
• Ok so we need information of how many of them are relative prime to each other? – mathreadler Mar 3 '15 at 21:49

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 $k=2$.