I keep arriving at a contradiction here, and I was hoping someone could point out my logic error, because it is pushing me to insanity....
Let $Q$ denote the rationals.
Consider $\alpha = \sqrt[4]{5}$. Then $\gamma := \alpha + i\alpha$ is a root of the irreducible $f(x) = x^4 + 20$.
Let $K = Q(\gamma)$, and we see that $[K:Q] = 4$.
Then we immediately have $K\subset Q(i,\alpha)$.
Since $\alpha + i\alpha\in K$, we obtain the following consequences:
$\begin{eqnarray*} i\alpha &=& \frac{(\alpha + i\alpha) - (\alpha - i\alpha)}{2}&\in& K\text{(this is the problem, because }\alpha - i\alpha\notin K)\\ \alpha &=& (\alpha + i\alpha) - i\alpha &\in& K\\ i &=& \frac{i\alpha}{\alpha} &\in& K \end{eqnarray*}$
Therefore $K$ contains $Q(i,\alpha)$ and thus we have equality.
That is, $Q(\alpha + i\alpha) = Q(i,\alpha)$.
But $\alpha$ is a root of the irreducible $x^4 - 5 = 0$, so $[Q(\alpha):Q] = 4$ and $i\notin Q(\alpha)$. Therefore $[Q(\alpha,i):Q(\alpha)] = 2$.
Then by the Tower law, $[K:Q] = [Q(i,\alpha):Q] = [Q(i,\alpha):Q(\alpha)][Q(\alpha):Q] = 8$.
So $[K:Q]$ obviously cannot be two different values. So which calculation is wrong?