Find not irreducible polynomial

Question

Actually I tried solving it.

A) By Eisenstein's Criteria its is irreducible.

B) and C) These polynomials don't have any zero in $\mathbb Z$ hence are also irreducible.

D) I have doubt in this tried it like ... Let $$x=y+1$$ then equation becomes $$y^4 + 5y^3+10y^2+10y+5.$$

By Eisenstein's Criterion this is also irreducible.

But its not possible, one of them is not irreducible!! Can someone help me?

Unrelated: I know only one Eisenstein criterion. Do you know another one? — Bernard, Jun 28 '17 at 23:49

Lukas Heger · Answer 1 · 2017-06-28T22:37:06.603

5

$2x^3+6x+12 =2(x^3+3x+6)$

Note that $2$ is not a unit in $\mathbb Z [x]$.

The criterion you cited (that having no roots implies irreducibility for polynomials of degree $\leq 3$) works only over fields.

edited Jun 28 '17 at 22:37

answered Jun 28 '17 at 22:35

Lukas Heger

1

Its over Z not Z[x] – user458361 Jun 28 '17 at 22:36
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@user458361 irreducible over $\mathbb Z$ is the same thing as being an irreducible element in the polynomial ring $\mathbb Z[x]$ – Lukas Heger Jun 28 '17 at 22:37
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@user458361 For elaboration on the prior comment see this recent question. – Bill Dubuque Jun 28 '17 at 22:46
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@MatheiBoulomenos Thanks a lot bro!! – user458361 Jun 29 '17 at 01:02

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