This question is from exercise in Chapter VI Q1 (n) in Lang's algebra.
Find the Galois group of $x^{4}-t$ over $\mathbb{R}(t)$, where $t$ is transcendental over the complex number $\mathbb{C}$
In fact I don't have any ideas about this question. I have finished Q1 (a) through (l), but I am stuck in here without any thoughts...
The point here is that I don't get any intuition about the transcendental element, and I think this question goes from here.
Any hints or explanations are highly appreciated!!!
EDIT 1:
Okay, for now, I learned something of the transcendental element. Since $t$ is a transcendental element over $\mathbb{C}$, then it cannot be the root of any polynomials with coefficients in $\mathbb{C}$.
However, if $t$ is transcendental over $\mathbb{C}$, it must be transcendental over $\mathbb{R}$.
I have figured out the following thing, I will keep updating.
Using the fact that $t$ is transcendental over $\Bbb{C}$ we can conclude that the polynomial is irreducible (I think that this is where they want you to go with this). Also, the roots of the polynomial must be distinct, so the extension is Galois.
Edit (additional explanation): Since $t$ is transcendental over $\Bbb{R}$, $\Bbb{R}[t]$ is isomorphic to a polynomial ring of $\Bbb{R}$ in one variable. So, $t$ is prime since it is irreducible. Then, by Eisenstein's criterion, $x^4-t$ is irreducible over the field of fractions of $\Bbb{R}[t]$ which is just $\Bbb{R}(t)$.
Call the splitting field of the polynomial $F$. Call the roots of the polynomial $\alpha_1,\cdots,\alpha_4$. In fact, if $\alpha$ is a root, then $-\alpha$ is a root. So we can just say that the roots are $\alpha_1,-\alpha_1,\alpha_2,-\alpha_2$. So $F=\Bbb{R}(t)(\alpha_1,\alpha_2)$. So $x^4-t=(x^2-\alpha_1^2)(x^2-\alpha_2^2)=x^4-(\alpha_1^2+\alpha_2^2)x^2+\alpha_1^2\alpha_2^2$. If $\alpha_2 \in \Bbb{R}(t)(\alpha_1)$, then $\alpha_2^2 \in \Bbb{R}(t)(\alpha_1)$, which would imply that there is some $a,b,c,d \in \Bbb{R}$ such that $\alpha_2^2 = a + b\alpha_1+c\alpha_1^2+d\alpha_1^3$. but this would imply that $\alpha_1^2 + a + b\alpha_1+c\alpha_1^2+d\alpha_1^3 =0$ contradicting that $x^4-t$ is the minimal polynomial for $\alpha_1$ over $\Bbb{R}(t)$. So we have two extensions which are $\Bbb{R}(t)(\alpha_1,\alpha_2)/\Bbb{R}(t)(\alpha_1)$ and $\Bbb{R}(t)(\alpha_1)/\Bbb{R}(t)$. Use this information to find the Galois group of the extension.