Transforming a Polynomial to Show Irreducibility Using Eisenstein's Criterion

Question

I have a particular polynomial

$$z^5-5z^4+30z^3-150z^2+465z-725$$

A quick check in mathematica shows that this polynomial is irreducible over the rationals, however, it does not pass the third condition of Eisenstein's Criteria to prove it definitively. Unfortunately, $$5^2|725$$

My question is this. I know you can make a transformation $z=x-c$ and once the transformed polynomial can be shown by EC to be irreducible then the original polynomial is irreducible. What is a suitable value for $c$ and is there a definitive or canonical approach to determining the possible value of $c$?

Try $c=1$. Then we obtain $x^5 - 10x^4 + 60x^3 - 280x^2 + 880x - 1376$. — Dietrich Burde, May 03 '16 at 12:42
@DietrichBurde As I understand he want backwards reasoning for the choice of c — Coolwater, May 03 '16 at 12:43
You can do even rational substitutions, see here. It does not always work, though. — Dietrich Burde, May 03 '16 at 12:45

score 2 · Answer 1 · edited Apr 13 '17 at 12:21

2

In general there is no suitable value for $c$. This has been shown at the answers to this MSE question, with the example $$ f(x) = x^3 + x + 1\in \mathbb{Z}[x]. $$

edited Apr 13 '17 at 12:21

Community

1

answered May 03 '16 at 12:48

Dietrich Burde

130,978

score 2 · Accepted Answer · answered May 03 '16 at 12:48

2

Hint: For a polynomial of $n$.degree and a coefficient $a_{n-1}\neq 0$. You can always reduce the $(n-1)$-power by $z=x+\frac{a_{n-1}}{n}$.

As Dietrich Burde suggested $c=1$ for this case.

answered May 03 '16 at 12:48

MrYouMath

15,833

Transforming a Polynomial to Show Irreducibility Using Eisenstein's Criterion

2 Answers2