Not that I fully understand quadratic integer rings yet, but I've been wondering about quartic integer rings.
Which leads me to the question: what kinds of algebraic integers generate quartic integer rings? I am fairly certain only square roots of squarefree integers are algebraic integers of degree $2$. So for degree $4$, these are the numbers that I am aware of:
- Fourth roots of squarefree integers with prime factors, like $\root 4 \of 5$, $\root 4 \of 6$, etc. (so $-i$ is not among these, despite being a fourth root of $1$).
- Sums of two square roots of coprime squarefree integers, like $\sqrt 2 + \sqrt 3$, $\sqrt 5 + \sqrt 7$.
What am I missing?