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This question already has an answer here:

Since Eisenstein´s criterion ist not directly applicable we look at the polynomial P (mod 5), which then reduces to $P(x) = x^n + 3$ (mod 5) ... is this correct? Now I can apply the Eisenstein Criterion with prime number 3: The leading coefficient of $x^n$ is 1 and not divisible by 3. The other coefficients, 0 and 3 are divisible by the prime 3. Further, the constant, 3, ist not divisible by the square of the chosen prime, $3^2 = 9$. Therefore, $P$ is an irreducible polynomial. Is this correct? Have I understood the E.C. correctly?

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marked as duplicate by lulu, Community Nov 4 '18 at 0:26

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  • $\begingroup$ Worth noting: I don't see how to use Eisenstein literally here, but a simple modification of the E.C. works fine, as is illustrated in the answers to the linked duplicate. $\endgroup$ – lulu Nov 3 '18 at 23:37
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No, this is not correct. Eisenstein's criterion is for polynomials in $\mathbb{Z}[x]$, not in $\mathbb{Z}_p[x]$, for some prime $p$. For instance, by your argument, $x^2+5x+6$ would be irreducible. But it is reducible, since it is equal to $(x+2)(x+3)$.

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  • $\begingroup$ Thank you - appreciated. I checked out on Eisenstein´s theorem and previous answers. $\endgroup$ – Parzifal Nov 4 '18 at 17:52
  • $\begingroup$ @Parzifal If my answer was useful, perhaps that you could mark it as the accepted one. $\endgroup$ – José Carlos Santos Nov 4 '18 at 17:54
  • $\begingroup$ your answer is marked. But if (and only if) you could spare a comment or two to explain how the Eisenstein criterion could be used for this task I would be more than grateful (I`m a high school student with not too much background). Thanks. $\endgroup$ – Parzifal Nov 5 '18 at 20:37
  • $\begingroup$ As much as I would love to help you, I see no way of using Eisenstein's criterion to solve this problem. $\endgroup$ – José Carlos Santos Nov 5 '18 at 21:02
  • $\begingroup$ Thank you anyway ... I was asked to use by my tutor to apply it, but I appreciate that it mustbe solved otherwise. $\endgroup$ – Parzifal Nov 7 '18 at 7:25

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