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I have this polynomial: $$ x^4 - 11x^3 + 27x^2 - 11x -13$$ And I need to check that is irreducible in $\mathbb{Q}[x]$. What I saw is that if it has a factorisation 3-1, it happens that must have a root that must be one of $\pm 1, \pm 13$, and any of them is. For the case of 2-2 I'm not able to see an easy way to attack it. I have a hint that says evaluate $f(x+a)$ for an adecuate $ a $ and then use the Einsenstein criterion to finish but I don't know how to get this appropiate $ a $, and why if I find it, it would finish the problem. Can anyone explain me?

iam_agf
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Note $\,f\,$ has discriminant $\,\color{#c00}7^3 11^2 19\,$ and $\ f(x) \equiv (x-1)^4 \pmod{\color{#c00} 7}\ $ is a prime power.

Thus $\,f(x+1) = x^4-7x^3+14x-7 \equiv x^4\pmod{7}\ $ does the trick. Below is why.

Hint $\ $ Recall that the key idea behind the Eisenstein criteria is that polynomials satisfying the criterion are, mod $\,p,\,$ powers of a prime, viz. $\,\equiv x^n,\,$ and products of primes always factor uniquely. The same works for its shift $\,(x-c)^n,\,$ so we seek primes $\,p\,$ such that, mod $\,p,\,$ the polynomial is congruent to such a power (e.g. for motivation: cyclotomic case). One can show that the only primes $\,p\,$ that can yield such powers are those dividing the discriminant of the polynomial. Indeed, if $\,f\equiv a (x-c)^n,\,\ n> 1\,$ then $\,f\,$ and $\,f'\,$ have a common root $\,x\equiv c,\,$ hence their resultant $\, R(f,f')\equiv 0.\,$ But this is, up to sign, the discriminant of $\,f\,$ (presumed monic).

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Bill Dubuque
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  • Why you know that $ a=-1$ is the useful one? (I mean, how you found it quickly? Does exist an algorithm or a rule?) – iam_agf Nov 03 '16 at 03:34
  • @MonsieurGalois That's explained in the Hint. – Bill Dubuque Nov 03 '16 at 03:37
  • Sorry, I'm still reading it. – iam_agf Nov 03 '16 at 03:38
  • As I understood, the key idea of the Einsenstein criteria is sending the polynomials to $\mathbb{Z}/p^2$ to get if it's reducible there, no? So if you get a polynomial there, you can change the variable $ x $ to $ x-c $ and check reducibility of it. This is the reason of why is useful the Einsenstein criteria in this case. Is what you're saying? – iam_agf Nov 03 '16 at 03:50