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
Hurkyl
Aug17-08, 10:31 PM
There aren't many solutions to bd = -2....
Niall101
Aug17-08, 10:46 PM
hey thanks for your reply I managed to get it. It is reducable. :)
Niall101
Aug17-08, 10:52 PM
Sorry for another question. If i have F(x)=x^3-5 Is it sufficiant to show it have no linear factors by plugging in +-5 +-1 and showing not equal to zero? I was thinking because the only factors could be linear or of degree 3. Thanks again.
Hurkyl
Aug17-08, 11:05 PM
The only limit on the ways that polynomials can factor is that the degrees of the factors add up to the degree of the polynomial. In particular, there are cubics that factor into a linear and a quadratic.
Niall101
Aug17-08, 11:11 PM
Over Q tho are +-5 +-1 The only possible roots of x^3-5?
cool thanks so much!
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 2 +A +A^2?
I really appreciate the help!
Edit: I like your site with the comics made me smile
Hurkyl
Aug17-08, 11:28 PM
The method you suggest sounds like you're planning on using linear algebra to compute the minimal polynomial of your field element. That will work, but I doubt it's the most efficient way to do it by hand.
Niall101
Aug17-08, 11:31 PM
Yeah i think thats what i was trying to do. Have to do this in an exam later. But We havent covered minimal polynomials yet so he probably wants a different method. Could you point me in the right direction?
Hurkyl
Aug17-08, 11:33 PM
What about the direct approach? Write down an indeterminate field element, and solve the equation that says your field element times the general field element equals 1.