I want to prove that there are infinitely many values of k such that the polynomial $x^{9}+12x^{5}-21x+k$ is irreducible. I sense that I have to use Eisenstein and the number 3 but I don't see exactly how. Any help would be appreciated.
Asked
Active
Viewed 139 times
1 Answers
1
General idea: What does Eisenstein with the prime $3$ say if $k=3$? What about $k=6$? What about $k=9$? Can you now find infinitely many $k$ that work?
Arthur
- 199,419
-
1Ok so the idea is to take k to be all numbers that are multiples of 3 but not divisible by 9. The Eisenstein for p=3 works every time right? – Teplotaxl Aug 02 '20 at 00:18
-
@Tepotaxl Yup. That's it. – Arthur Aug 02 '20 at 04:14