# Irreduclible polynomials

• Apr 13th 2009, 04:52 AM
ZetaX
Irreduclible polynomials
Show that $f(x)=x^4+bx^2+d$ is irreducible over
$\mathbb Q(\sqrt{d(b^2-4d)})[x]$
• Apr 13th 2009, 01:04 PM
NonCommAlg
Quote:

Originally Posted by ZetaX

Show that $f(x)=x^4+bx^2+d$ is irreducible over $\mathbb Q(\sqrt{d(b^2-4d)})[x]$

this can't be right! did you mean reducible?
• Apr 13th 2009, 03:32 PM
ZetaX
No this is correct, it is irreducible, that is what I am suppose to show.
• Apr 13th 2009, 04:08 PM
NonCommAlg
Quote:

Originally Posted by ZetaX

No this is correct, it is irreducible, that is what I am suppose to show.

the question doesn't make sense! for example, choose d = 1 and b = 3. then $\sqrt{d(b^2 - 4d)}=\sqrt{5}.$ now in $\mathbb{Q}(\sqrt{5})[x]$ we have:

$x^4+3x^2 + 1 = \left(x^2 + \frac{3 + \sqrt{5}}{2} \right) \left(x^2 + \frac{3 - \sqrt{5}}{2} \right).$
• Apr 14th 2009, 06:47 AM
ZetaX
I have that
$
f(x)=x^4+ax^2+b
$

is irreducible over rationals, and its galois group is
$
D_4
$

And M is the splitting field of $f(x)$

and I have that $

\mathbb Q(\sqrt{d(b^2-4d)})
$

is a quadratic subfield of M.
• Apr 14th 2009, 12:07 PM
ThePerfectHacker
Quote:

Originally Posted by NonCommAlg
this can't be right! did you mean reducible?

Quote:

Originally Posted by ZetaX
No this is correct, it is irreducible, that is what I am suppose to show.

Quote:

Originally Posted by NonCommAlg
the question doesn't make sense! for example, choose d = 1 and b = 3. then $\sqrt{d(b^2 - 4d)}=\sqrt{5}.$ now in $\mathbb{Q}(\sqrt{5})[x]$ we have:

$x^4+3x^2 + 1 = \left(x^2 + \frac{3 + \sqrt{5}}{2} \right) \left(x^2 + \frac{3 - \sqrt{5}}{2} \right).$

Quote:

Originally Posted by ZetaX
I have that
$
f(x)=x^4+ax^2+b
$

is irreducible over rationals, and its galois group is
$
D_4
$

And M is the splitting field of $f(x)$

and I have that $

\mathbb Q(\sqrt{d(b^2-4d)})
$

is a quadratic subfield of M.

I think that ZetaX is referring to this thread. In that thread he asks that given that $x^4 + bx^2 + d$ is irreducible over $\mathbb{Q}$ with Galois group $D_4$ then show that the three quadradic extensions (we know there has to be three since $D_4$ has three subgroups of index $2$) are: $\mathbb{Q}(\sqrt{b^2-4d}),\mathbb{Q}(\sqrt{d}),\mathbb{Q}(\sqrt{d(b^2-4d)})$. He is asking to complete that solution.
• Apr 14th 2009, 12:23 PM
ZetaX
I am not asking to complete this but the question I am asking is continuation of this.