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Is $f(X)=(2+i)X^4+5X^2-3i$ irreducible over $\mathbb{Z}[i]$?

I would be pleased if you could tell me how to start.

Thanks.

Bill Dubuque
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maiT
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1 Answers1

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Hint $ $ Apply Eisenstein to the reverse ("reciprocal") polynomial $\, x^4 f(x^{-1})\,$ using $\,p = 2+i$.

Note that the reversal map $\, f\mapsto x^d f(x^{-1}),\ d=\deg f\,$ is multiplicative, being the product of two multiplicative maps, namely $\,f(x)\mapsto f(x^{-1}),\,$ and $\,f(x)\mapsto x^{\,\deg f}\ $ (an "exponential" of the additive degree map). Being multiplicative, it preserves (ir)reducibility (trivial exception when $\,x\mid f\,$ doesn't arise here [or in practice] since then irred. testing is trivial - no need to reverse).

Bill Dubuque
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  • Thank you. I tried it and showed with Eisenstein that $x^4*f(x^{-1})$ is irreducible but why is $f(x)$ irreducible? – maiT Dec 06 '16 at 20:43
  • @milui I added a note about that. – Bill Dubuque Dec 06 '16 at 20:58
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    Multiplicativity does not guaranteed that irreducibility of the reciprocal is equivalent to irreduciibility of the polynomial we started with. What is missing here is that going reciprocal is its own inverse. For a full proof see these older threads 1, 2. The problem appears only with the single irreducible polynomial with a vanishing constant term. – Jyrki Lahtonen Aug 05 '25 at 08:00
  • Prior comment is irrelevant in OP (and in general), see my edit $\ \ $ – Bill Dubuque Aug 05 '25 at 20:42