# IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q?

1. Aug 17, 2008

### Niall101

IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

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
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!

Niall

2. Aug 17, 2008

### Hurkyl

Staff Emeritus
Re: IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

There aren't many solutions to bd = -2....

3. Aug 17, 2008

### Niall101

Re: IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

hey thanks for your reply I managed to get it. It is reducable. :)

4. Aug 17, 2008

### Niall101

Re: IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

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.

5. Aug 17, 2008

### Hurkyl

Staff Emeritus
Re: IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

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.

6. Aug 17, 2008

### Niall101

Re: IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

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!

Last edited: Aug 17, 2008
7. Aug 17, 2008

### Hurkyl

Staff Emeritus
Re: IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

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.

8. Aug 17, 2008

### Niall101

Re: IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

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?

9. Aug 17, 2008

### Hurkyl

Staff Emeritus
Re: IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

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.

10. Aug 18, 2008

### Niall101

Re: IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

great thanks I got it now. Thanks for your help!!