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

Sure: take $g$ to be the minimal polynomial of $a^n$. – Crostul Jul 04 '15 at 23:53 — Crostul, Jul 04 '15 at 23:53

Khadija Mbarki · Answer 1 · 2015-07-05T00:03:58.970

3

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 $a^n.$

edited Jul 05 '15 at 00:03

answered Jul 04 '15 at 23:56

Khadija Mbarki

631

Could you kindly give examples of polynomials that are not linear? Thanks! – Jul 05 '15 at 00:16
These polynomials are not linears – Khadija Mbarki Jul 05 '15 at 00:17
They are. They are of degree 1, right? – Jul 05 '15 at 00:36
Yes but as i know a linear polynomial is of the form $\alpha x.$ – Khadija Mbarki Jul 05 '15 at 07:18

score 0 · Answer 2 · answered Jul 05 '15 at 01:20

0

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 $x^4-10x^2+1$.

answered Jul 05 '15 at 01:20

Lucian

48,334
2
83
154

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

2 Answers2