TITLE: $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational
QUESTION [9 upvotes]: Prove that $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational.

I would assume that $\sqrt[31]{12} +\sqrt[12]{31}$ is rational and try to find a contradiction.
However, I don't know where to start. Can someone give me a tip on how to approach this problem?

REPLY [7 votes]: It is known that algebraic integers are closed under addition, subtraction, product and taking roots.
Since $12$ and $31$ are algebraic integers, so does their roots $\sqrt[31]{12}$, $\sqrt[12]{31}$. Being the sum of two such roots, $\sqrt[31]{12} + \sqrt[12]{31}$ is an algebraic integer.
It is also known that if an algebraic integer is a rational number, it will be an ordinary integer. Notice
$$2 < \sqrt[31]{12} + \sqrt[12]{31}
< \sqrt[31]{2^4} + \sqrt[12]{2^5} = 2^{\frac{4}{31}} + 2^{\frac{5}{12}} < 2\sqrt{2} < 3$$
$\sqrt[31]{12} + \sqrt[12]{31}$ isn't an integer and hence is an irrational number.