# Question on the algebraic structure of the set $\mathcal{P} = \{ \sum_{i=1}^n a_i x^i = 0, a_i \in \pm \mathbb{P}~or~0\}$

It is well-known that the set $$\mathcal{A} = \bigg\{ x\in \Bbb C: \sum_{i=1}^n a_i x^i = 0, a_i \in \mathbb{Z}\text{ and } n \text{ is a positive integer}\bigg\},$$ is the set of all algebraic numbers, so by a heuristic argument, now let's consider the set $$\mathcal{P} = \bigg\{ x\in \Bbb C : \sum_{i=1}^n a_i x^i = 0, a_i \in \pm \mathbb{P}\cup\{ 0\}\text{ and }n\text{ is a positive integer}\bigg \},$$ where $$\mathbb{P}$$ is the set of all prime numbers. Now the question is: does the set $$\mathcal{P}$$ has a good algebraic structure?

• By "algebraic structure" I guess you mean to say that the first set $\mathcal{A}$ is also a field. The operations on $\mathcal{A}$ transfer computatively to the $a_i$'s. Hence you can check whether $\mathcal{P}$ has an algebraic structure (group, ring, ...) by hand. Or, more likely, you can compute explicit counter-examples to any such structure induced by the usual operations. Dec 28 '20 at 10:36
• Out of curiosity, is there a particular motivation behind considering $\mathcal{P}$? Dec 28 '20 at 10:56
• Can you prove $\mathcal P$ is not the set of all algebraic numbers? For example is $\sqrt{6} \in \mathcal P$? Dec 28 '20 at 12:59