# Eisenstein Criterium

• Sep 28th 2008, 12:18 PM
roporte
Eisenstein Criterium
Investigate if f is irreducible in $Q[X]$ and $Z[X]$:

i) $f=2X^5-6X^3+9X^2-15$
ii) $f=3X^4+6X^2+6$

Thanks!!!
• Sep 28th 2008, 12:42 PM
ThePerfectHacker
Quote:

Originally Posted by roporte
i) $f=2X^5-6X^3+9X^2-15$
ii) $f=3X^4+6X^2+6$

Irreducibility in Z and Q are equivalent by Gauss' lemma. Now in (i) let p=3 then p divides -6,9,-15 by p^2 does not divide -15 and so it is irreducible. In (ii) let p=2 then p divides 6 and 6 but p^2 does not divide 6.