TITLE: What kinds of algebraic integers are of degree $4$?
QUESTION [8 upvotes]: 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?

REPLY [3 votes]: I choose not to be daunted by this question, even though in some regards it is daunting, however much that has been overplayed so far. I think that a lot of these algebraic integers of degree $4$ can be boiled down to $a + b \theta + c \theta^2 + d \theta^3$, where $a, b, c, d$ are all integers, or perhaps all rational numbers satisfying a certain  condition, and $\theta$ is an algebraic integer of degree $4$.
Something tells me that it is this $\theta$ that you're actually interested in, namely, your apparent ignorance of quadratic integers like $\omega$ and $\phi$ despite your earlier demonstrated acquaintance with them.
It is for these $\theta$ that things get hairy. What I have pieced together so far, mainly from your question and from comments:

Fourth roots of integers, provided they are not perfect powers and not divisible by any fourth powers.
Sums of two square roots of coprime squarefree integers.
Square root of an integer plus a square root.
Maybe the an integer plus a square root, divided by a square root or the cube of a square root?

At least we don't have to worry about Abel's impossibility theorem at this degree.