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Let p be a prime number. Question is for any prime value of p the polynomial $$1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...++\frac{x^p}{p!}$$ is irreducable. I found out that it seems like a taylor series but I couldnt solve the question.

Demir Eken
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  • I dont think that I understand your point. Can you explain your solution more detailed? – Demir Eken Jul 18 '18 at 07:35
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    This is actually irreducible for all integer exponents, but for non-prime it is harder to prove. If you are interested look for Irreducibility of truncated exponentials. – Sil Jul 18 '18 at 07:43
  • Duplicate (target) of https://math.stackexchange.com/questions/3947268/truncated-expresion-of-ex-is-an-irreducible-polynomial – Martin Brandenburg Feb 15 '25 at 12:28

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Multiply by $p!$, and apply the Schönemann-Eisenstein theorem with the prime $p$.

A. Pongrácz
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