Can two irreducible polynomials have different powers of the same real number as roots?

Question

Say we have two irreducible polynomials in $Q [x] $. We call them $f, g$. Say one of the roots of $f $ is $a$. Is it possible that $g$ satisfies a root of the form $a^n$ for some natural number $n $?

Thanks

Answer 1

Take $f(x)=x-a$ and for a fixed naturel number $n,$ $g(x)=x-a^n.$ The two polynomials are irreducibles in $Q[x]$ and the root of $f$ is $a$ and for $g$ is $a^n.$

Answer 2

Hint: Take a root of the form $A\sqrt\alpha+B\sqrt\beta$, where both a and b are square-free. For instance, for $A=B=1$, $\alpha=2$, $\beta=3$, and $n=2$, we have $x^2-10x+1$ and $x^4-10x^2+1$.