True or False statements of Splitting Fields

Question

Verify if the following statements is true or false:

A: The splitting field of $X^2-X-1$ is equal to $\mathbb{Q}[\sqrt{5}]$

B: The splitting field of $X^4-5$ is equal to $\mathbb{Q}[\sqrt[4]{5}]$

What have I tried so far?

I have factorised A to give the splitting field $\mathbb{Q}[\frac{1+\sqrt{5}}{2},\frac{1-\sqrt{5}}{2}]$

And for B we get $\sqrt[4]{5}$

Hence I think A is false and B is true.

Answer 1

For A: $\Bbb Q[\sqrt 5] = \Bbb Q[\frac{1+\sqrt 5}{2}, \frac{1-\sqrt 5}{2}]$, since $\sqrt 5$ is in $\Bbb Q[\frac{1+\sqrt 5}{2}, \frac{1-\sqrt 5}{2}]$ and $\frac{1\pm\sqrt 5}{2}$ are in $\Bbb Q[\sqrt 5]$.

Answer 2

For B: note that $X=i\sqrt[4]5$ satisfies $X^4-5=0$, but $i\sqrt[4]5\not\in \Bbb Q(\sqrt[4]5)\subset\mathbb R$.