Is $f_d(X, Y) =X^d+Y^d+1$ is irreducible as a polynomial over complex numbers for $d>1$ and $d$ natural? If not, there exists a irreducible polynomial of any degree? Because that is what I'm looking for...
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Consider $d=1$, then $f(x,y)=x+y+1$, is irreducible. The condition should be $d>1$. – Hector Blandin Nov 21 '17 at 19:35
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1Have you reached a conclusion for any small degrees $d$? – hardmath Nov 21 '17 at 19:41
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Hey! I don't see how the first part of my question is related with the $X^n-Y^m$ question. It's a particular case that i don't see? Otherwise it seems a bit unfair to flag my question as a duplicate – Hurjui Ionut Nov 22 '17 at 09:19
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I only tried to show for small $d$ by brute force (i.e suppose that $f$ can be written as a product of two polynomials). I not an algebra specialist so that im asking for a more elegant solution that i can learn from. – Hurjui Ionut Nov 22 '17 at 09:24
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@Hector $d \geq 1$ * – Hurjui Ionut Nov 22 '17 at 09:25
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Oh Yes, exactly – Hector Blandin Nov 22 '17 at 12:15