While learning Galois theory, I tried to construct a splitting field for the polynomial $x^4+2$ over $\mathbb{Q}$, but I am terribly stuck.
Since $x^4+2$ is irreducible by Eisenstein's criterion, I started by setting $K=\mathbb{Q}[y]/(y^4+2)$. Over this field I can factor $x^4+2=(x+y)(x-y)(x^2+y^2)$, but I was not able to find out wether the last factor is irreducible in $K[Y]$ or not.
(Hint: SAGE tells me it is irreducible.)
So I was send on a journey to learn how to find out such things. I was able to figure out the ring of integers of $K$ is $\mathcal{O}_K=\mathbb{Z}[y]/(y^4+2)$ via some trial and error with norms, but don't know how to proceed, mainly because I have trouble determining the prime ideals in $\mathcal{O}_K$, and have yet to gain a better understanding of how this knowledge would help (something valuations and Newton polygons).
If possible, I would like to avoid proving that $x^2+y^2$ has no zeroes by "brute force", i.e. taking a general element of $K$ and finding a contradiction. Is there some more constructive way to see this, either geometrically or algebraically?
A reference which describes how to do such things, with examples, would also be very welcome.
Just fix $\alpha\in \mathbb{C}$ some root of the polynomial. If $u$ is another solution, then $(\frac{u}{\alpha})^4=1$, so $u$ is solution of the equation $t^4-\alpha^4=0$. Then $u\in\{\alpha,-\alpha,\alpha\cdot i, -\alpha\cdot i\}$. So the splitting field is just $\mathbb{Q}(\alpha,i)$. As you can see you don't even need to know the explicit form of any root