To calculate $[\mathbb Q(\sum_{j=1}^ka_j^{1/n_j}) : \mathbb Q]$ , when $a_i , n_i$ are positive integers with $a_1^{1/n_1}$ not an integer

Question

Let $a_1 , n_1$ be positive integers such that $a_1^{1/n_1}$ is not an integer . Then I can show that for any positive integers $a_2,...,a_k,n_2,...,n_k$ , $\sum_{j=1}^ka_j^{1/n_j}$ is an irrational number .

My question is , can I determine the extension degree $[\mathbb Q(\sum_{j=1}^ka_j^{1/n_j}) : \mathbb Q]$ ? Or at least , is it possible to give a good upper and lower bounds ( I know it is bounded below by $2$) ?

Answer 1

You cannot be guaranteed a lower bound greater than 2. For instance let $a_i = 2^{i}$ and $n_i = 2i$

Then each $a_i^{\frac{1}{n_i}} = (2^i)^{\frac{1}{2i}} = 2^{\frac{1}{2}}= \sqrt2$