Problem

Source: Iran 3rd round 2012-Algebra exam-P3

Tags: algebra, polynomial, algebra proposed



Suppose $p$ is a prime number and $a,b,c \in \mathbb Q^+$ are rational numbers; a) Prove that $\mathbb Q(\sqrt[p]{a}+\sqrt[p]{b})=\mathbb Q(\sqrt[p]{a},\sqrt[p]{b})$. b) If $\sqrt[p]{b} \in \mathbb Q(\sqrt[p]{a})$, prove that for a nonnegative integer $k$ we have $\sqrt[p]{\frac{b}{a^k}}\in \mathbb Q$. c) If $\sqrt[p]{a}+\sqrt[p]{b}+\sqrt[p]{c} \in \mathbb Q$, then prove that numbers $\sqrt[p]{a},\sqrt[p]{b}$ and $\sqrt[p]{c}$ are rational.