- #1
roam
- 1,271
- 12
I have a question about the application of Eisenstein’s Criterion. I want to show that
g(x)=x4+4x3+7x+5 is irreducible over [tex]\mathbb{Q}[/tex].
That 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) = x4 + 8x3 + 18 x2 + 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?
g(x)=x4+4x3+7x+5 is irreducible over [tex]\mathbb{Q}[/tex].
That 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) = x4 + 8x3 + 18 x2 + 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?