# splitting field L

• Mar 22nd 2009, 10:41 PM
dopi
splitting field L
B is a real number Sqrt(7 + 2*sqrt(7))

basically i have done the parts of finding the minimal polynomial mu of B over Q, and all the zeros of mu.

but i got stuck on the part where i need to find the degree [L :Q] and the basis for L over Q

Where L is the splitting field of mu over Q

can anyone help me find the degree and basis

thanks
• Mar 23rd 2009, 07:53 PM
ThePerfectHacker
Quote:

Originally Posted by dopi
B is a real number Sqrt(7 + 2*sqrt(7))

basically i have done the parts of finding the minimal polynomial mu of B over Q, and all the zeros of mu.

but i got stuck on the part where i need to find the degree [L :Q] and the basis for L over Q

Where L is the splitting field of mu over Q

can anyone help me find the degree and basis

thanks

Let $\displaystyle x = \sqrt{7+2\sqrt{7}} \implies x^2 - 7 = 2\sqrt{7} \implies x^4 - 14x^2+21=0$.
Find the zeros of this polynomial and form the splitting field.
• Mar 24th 2009, 01:39 PM
dopi
check solution
Quote:

Originally Posted by ThePerfectHacker
Let $\displaystyle x = \sqrt{7+2\sqrt{7}} \implies x^2 - 7 = 2\sqrt{7} \implies x^4 - 14x^2+21=0$.
Find the zeros of this polynomial and form the splitting field.

i found the zeros as

$\displaystyle \beta = \pm \sqrt{7 \pm 2\sqrt{7}}$
$\displaystyle \alpha= \pm \sqrt{7 \mp 2\sqrt{7}}$

so i tried $\displaystyle \alpha\beta$, where i got $\displaystyle \sqrt{21}$, so is the splitting field L of mu over Q = $\displaystyle \sqrt{7}$.

is the degree[L,:Q] = 4x4 = 16
and the basis for L over Q {$\displaystyle {1, \sqrt{7},7, 7\sqrt{7}}$}
and the order of the galois group $\displaystyle \Gamma(L:Q) = 8$
• Mar 24th 2009, 07:15 PM
ThePerfectHacker
Quote:

Originally Posted by dopi
i found the zeros as

$\displaystyle \beta = \pm \sqrt{7 \pm 2\sqrt{7}}$
$\displaystyle \alpha= \pm \sqrt{7 \mp 2\sqrt{7}}$

so i tried $\displaystyle \alpha\beta$, where i got $\displaystyle \sqrt{21}$, so is the splitting field L of mu over Q = $\displaystyle \sqrt{7}$.

is the degree[L,:Q] = 4x4 = 16
and the basis for L over Q {$\displaystyle {1, \sqrt{7},7, 7\sqrt{7}}$}
and the order of the galois group $\displaystyle \Gamma(L:Q) = 8$

The splitting field is $\displaystyle \mathbb{Q}(\sqrt{7 + 2\sqrt{7}},\sqrt{7-2\sqrt{7}})$.
• Mar 24th 2009, 07:26 PM
dopi
is the degree[L,:Q] = 4x4 = 16
and the basis for L over Q {$\displaystyle {1, \sqrt{7},7, 7\sqrt{7}}$}
and the order of the galois group $\displaystyle \Gamma(L:Q) = 8$[/QUOTE]

i was wondering are these correct?