- #1

roam

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**g(x)=x**is irreducible over [tex]\mathbb{Q}[/tex].

^{4}+4x^{3}+7x+5That means I need to find a prime number p such that

[tex]p \nmid 1[/tex] , [tex]p | 4[/tex] , [tex]p | 7[/tex], [tex]p | 5[/tex] and [tex]p^2 \nmid 5[/tex].

But unfortunately I can't see any prime number which would satisfies this!

I

*think*there is a theorem that says if g(x+1) is irreducible then g(x) is irreducible. So in this case

g(x+1) = x

^{4}+ 8x

^{3}+ 18 x

^{2}+ 16 x + 17

But again I cannot find a p to satisfiy Eisenstein’s irreducibility criterion... So why does the method fail? Then what other method can one use to establish g(x)'s irreducibility?