What kinds of algebraic integers are of degree $4$? 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?
 A: Among other things, you are missing $~2x~=~\sqrt{\alpha-\beta}~-~\sqrt{\beta-\alpha+\dfrac2{\sqrt{\alpha-\beta}}}~,~$ which, for $~\alpha=\sqrt[3]{\dfrac12+\dfrac{\sqrt{849}}{18}}~$ and $~\beta=4~\sqrt[3]{\dfrac2{3(9+\sqrt{849})}}~,~$ is one of the solutions of $x^4-x-1=0$. 
The point being, explicit formulas for algebraic integers of degree four can be very, very complicated, much more complicated than for degree two. 
$($To get another version of this number, type $x^4-x-1=0$ into Wolfram Alpha, and after it gives you a numerical solution, ask it for the "exact form".$)$
A: Try a nontrivial fifth root of unity, being a root of $(x^5-1)/(x-1)$.  It is ultimately hopeless and not a wise use of time to try to describe all algebraic integers of degree 4.
Your description of all algebraic integers of degree 2 is incomplete also. Try $(1+\sqrt{5})/2$. There are many more examples where that came from.
A: 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.
