Exercise. we have following extensions:
a) $\mathbb{Q(\sqrt(2),\sqrt(5))}:\mathbb{Q}$
b)$\mathbb{Q(\alpha)}:\mathbb{Q}$ where $\alpha=e^{{2\pi i}/3}$
c)$K:\mathbb{Q}$ where splitting field over $\mathbb{Q}$ for $t^4 -3t^2+4$.
Check that the corresponding extensions are normal. I know that normal extension if every irreducible polynomial f over $\mathbb{Q}$ that has at least one zero in extension and splits it.
But how to show it in this ex? I need some help, explanations. b) if I understand correctly, because this extension has second degree so it splitting field, so normal. c) said that splitting field, so and normal. what about a)?
Best Answer
In (a), $\;\Bbb Q(\sqrt2,\,\sqrt5)\;$ is the splitting field of $\;(x^2-2)(x^2-5)\in\Bbb Q[x]\;$ or, if you prefer, since we have that $\;\Bbb Q(\sqrt2,\,\sqrt5)=\Bbb Q(\sqrt2+\sqrt5)\;$ (why?), and rationalizing this sum of squares:
$$r=\sqrt2+\sqrt5\implies r^2-2\sqrt2\,r-3=0\implies r^4-6r^2+9=8r^2\implies\sqrt2+\sqrt5$$
is a root of $\;p(x)=x^4-14x^2+9\in\Bbb Q[x]\;$ , so $\;\Bbb Q(\sqrt2+\sqrt5)\;$ is the splitting field over the rationals of $\;p(x)\;$ and thus a normal extension of $\;\Bbb Q\;$ .