Can we write $-1$ as a sum of square in the splitting field of $x^4 + 4$ over $\Bbb{Q}$?

Question

Can we write $-1$ as a sum of square in the splitting field of $x^4 + 4$ over $\Bbb{Q}$?
If so, how can we prove it?

Answer 1

Observe that

$$x^4 + 4 = (x^2 - 2x + 2)(x^2 + 2x + 2) = (x - (1+i))(x-(1-i))(x+(1-i))(x+(1+i))$$

In particular, the roots of $x \mapsto x^4 + 4$ are: $1+i, 1-i, -1+i, -1-i$

So, each of these roots is in the splitting field over $\mathbb{Q}$, which one can now see is simply $\mathbb{Q}(i)$.

Thus, $i$ is an element of the splitting field, which does contain the sum of squares $i^2 + 0^2 = -1$ as remarked in an earlier answer.

Answer 2

One way to write $-1$ as the sum of two squares is $-1=i^2+0^2$. Therefore if $i$ is in the splitting field, then the answer is yes.

Note that this is not an if and only if statement.