galois Theory

**peteryellow** · February 18th 2009, 08:13 AM

Determine, if there are rational numbers c,d so that the polynomails x^4 -4x-1 and x^4+ax^2+b have splitting field over Q(rational numbers), If yes find a and b.

**peteryellow** · February 18th 2009, 08:23 AM

it should be same splitting field

soory I forgor to write

**ThePerfectHacker** · February 18th 2009, 05:28 PM

Originally Posted by peteryellow

Determine, if there are rational numbers c,d so that the polynomails x^4 -4x-1 and x^4+ax^2+b have splitting field over Q(rational numbers), If yes find a and b.

I think it is impossible. This is just an idea that I have that might work. We make a trivial observation, if they have the same splitting field then the polynomials definitely have the same Galois group. The polynomial $x^4 - 4x - 1$ happens to have Galous group equal to $S_4$ (this can be shown using what your favorite method is). Thus, we require splitting field for $x^4 + ax^2 + b$ to be a degree $24$ extension. Now, the equation $x^4 + ax^2 + b=0$ solves into $x = \pm \sqrt{ \frac{-a+r}{2}}, \pm \sqrt{ \frac{-a-r}{2}}$ where $r = \sqrt{a^2 - 4b}$ . Therefore, the splitting field is $K=\mathbb{Q} \left( \sqrt{ \frac{-a+r}{2}}, \sqrt{\frac{-a+r}{2}} \right)$ . We require that $[K:\mathbb{Q}]=24$ . However, $[K:\mathbb{Q}] \leq \left[\mathbb{Q}\left( \sqrt{\frac{-a+r}{2}} \right):\mathbb{Q} \right]\left[ \mathbb{Q}\left( \sqrt{\frac{-a-r}{2}} \right) : \mathbb{Q} \right]$ . But $\left[\mathbb{Q}\left( \sqrt{\frac{-a+r}{2}} \right):\mathbb{Q} \right] \leq 4$ because $\sqrt{\frac{-a+r}{2}}$ solves $r^2 = (2x^2 + a)^2$ . Likewise, $\left[\mathbb{Q}\left( \sqrt{\frac{-a+r}{2}} \right):\mathbb{Q} \right] \leq 4$ . Therefore, $[K:\mathbb{Q}] \leq 16 < 24$ and so it seems it is impossible.

**ThePerfectHacker** · February 19th 2009, 09:20 AM

I think it is a good exercise to actually find the conditions on $a,b$ that determine what Galois group $x^4 + ax^2 + b$ actually is.
This should not be too bad since it appears as an exercise problem in my book.

**peteryellow** · February 22nd 2009, 04:23 AM

which book are you using?

**ThePerfectHacker** · February 22nd 2009, 06:48 AM

Originally Posted by peteryellow

which book are you using?

This one.

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