is f irreducible

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May 25th 2009, 09:37 AM
georgel

is f irreducible

Is $f(x)=x^3+15x^2+8x+6$ irreducible in $\mathbb{Z}[x]$ , $\mathbb{Z}[[x]]$ , $\mathbb{Z}_5[x]$ ?

I've tried doing the $\mathbb{Z}[x]$ , but we've only mentioned Eisenstein's criterion and I can't apply it here.

Thank you.
May 25th 2009, 10:33 AM
NonCommAlg

Quote:

Originally Posted by georgel

Is $f(x)=x^3+15x^2+8x+6$ irreducible in $\mathbb{Z}[x]$ , $\mathbb{Z}[[x]]$ , $\mathbb{Z}_5[x]$ ?

I've tried doing the $\mathbb{Z}[x]$ , but we've only mentioned Eisenstein's criterion and I can't apply it here.

Thank you.

the degree of $f$ is 3. so it's irreducible over $\mathbb{Z}$ iff it has no integer root. now suppose $x^3+15x^2+8x+6=0.$ then $x(x^2+15x+8)=-6$ and thus if $x \in \mathbb{Z},$ then we must have

$x=\pm 1, \pm 2, \pm 3, \pm 6.$ check these cases! the polynomial is reducible in $\mathbb{Z}[[x]]$ because for example: $f(x)=(1-x)(6+14x+29x^2 + 30x^3 + 30x^4 + 30x^5 + \cdots ).$

finally since $f(1) \equiv 0 \mod 5,$ we have $x-1 \mid f(x)$ in $\mathbb{Z}_5[x]$ and thus $f$ is reducible over $\mathbb{Z}_5.$
May 25th 2009, 11:12 AM
georgel

Quote:

Originally Posted by NonCommAlg

the degree of $f$ is odd. so it's irreducible over $\mathbb{Z}$ iff it has no integer root.

I didn't know this. Could you possibly link the theorem and its proof, if it's available somewhere online?
Maybe it's obvious but I don't see it. :(

And thank you so much, again!
May 25th 2009, 11:41 AM
georgel

Quote:

Originally Posted by NonCommAlg

the polynomial is reducible in $\mathbb{Z}[[x]]$ because for example: $f(x)=(1-x)(6+14x+29x^2 + 30x^3 + 30x^4 + 30x^5 + \cdots ).$

And I need further help with this as well. :( How did you get this above?

Thank you if you have the time to help, if not, I'm already grateful for all your help!
May 25th 2009, 11:50 AM
NonCommAlg

Quote:

Originally Posted by georgel

I didn't know this. Could you possibly link the theorem and its proof, if it's available somewhere online?
Maybe it's obvious but I don't see it. :(

And thank you so much, again!

sorry, i meant 3 not "odd". i don't know why i wrote odd?? well, there are only two ways to factorize a polynomial of degree 3: all factors of degree 1 or one degree 1 and one degree 2.

so either way, there will be a factor of degree 1 and therefore a root.