Do there exist $2k+1$ irrational numbers such that their product and sum are both rational?

I recently found a problem saying "Find 2 irrational numbers such that their sum and product is both rational."

After a while I noticed any pair like $$(a+\sqrt{b},a-\sqrt{b})$$ work .From this I could easily say this statement is true for any even integer.But then I thought whether the same is true for odd integers. I realized proving it for $$3$$ would imply the statement being true for all odd integers. I tried to work with similar expression as above.I also tried to work with the cubic polynomial.But I couldn't make any significant progress.

• Take any polynomial of odd degree with no rational roots, like $x^3-5x^2+3x+2$. That one has real roots (not sure you wanted to require that). the product of the roots is $-2$ and the sum is $5$. A similar approach works for any degree. – lulu May 30 at 19:35
• How to make sure that all its roots are real? – Yes it's me May 30 at 19:37
• I just adjusted the constant to get something that works. – lulu May 30 at 19:42
• To prove existence in general, it's probably easiest to use some Galois theory. See, e.g., this question. – lulu May 30 at 19:44

You could take $$\{j\:\sqrt[2k+1]{2}|1\le j\le 2k\}\cup\{-k(2k+1)\:\sqrt[2k+1]{2}\}$$. The sum is $$0$$; the product is $$-(2k+1)!2k$$.