IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??
Attempt:
Ive shown that this has no linear factors as +-1 +-2 are not roots.
Then I let F(x)=(x^2+ax+b)(x^2+cx+d)
And I end up with
a +c=1
d+ac+b=-1
ad+bc=-2
bd=-2
But i can seam to show why no a b c d exist (or Exist)
Is there an easier approach?
I have an exam in 12 hours so I hope soeone can help!
Thanks in advance!
Niall
i will note this theorem.
If is reducible over , then is reducible over .
so, as a contrapositive of the statement,
If is irreducible over , then is irreducible over .
from this,
then WLOG, and .
try to solve the combinations..
EDIT: aww, didn't notice that there were replies already..
If anyone has a chance to shed some light on this one it would be a great help
Let A = cuberoot(5),
and let
F = a + bA + cA in R
a; b; c in Q
Find the multiplicative inverse of the element 2 +A +A^2 in the field F.
For this I Know There is an element in the field which multiplied by this give the Multiplicitive identity. Is there a way of doing this by taking powers of A?