Number theorists study a range of different questions that are loosely inspired by questions related to integers and rational numbers.
Here are some basic topics:
Distribution of primes: The archetypal result here is the prime number theorem,
stating that the number of primes $\leq x$ is asymptotically $x/\log x$. Another basic result is Dirichlet's theorem on primes in arithmetic progression. More recently, one has the results of Ben Green and Terry Tao on solving linear equations (with $\mathbb Z$-coefficients, say) in primes. Important open problems are Goldbach's conjecture, the twin prime conjecture, and questions about solving non-linear equations in primes (e.g. are there infinitely many primes of the form $n^2 + 1$). The Riemann hypothesis (one of the Clay Institute's Millennium Problems) also fits in here.
Diophantine equations: The basic problem here is to solve polynomial equations (e.g. with $\mathbb Z$-coefficients) in integers or rational numbers.
One famous problem here is Fermat's Last Theorem (finally solved by Wiles). The theory of elliptic curves over $\mathbb Q$ fits in here. The Birch-Swinnerton-Dyer conjecture (another one of the Clay Institute's Millennium Problems) is a famous open problem about elliptic curves. Mordell's conjecture, proved by Faltings (for which he got the Fields medal) is a famous result. One can also study Diophantine equations mod $p$ (for a prime $p$). The Weil conjectures were a famous problem related to this latter topic, and both Grothendieck and Deligne received Fields medals in part for their work on proving the Weil conjectures.
Reciprocity laws: The law of quadratic reciprocity is the beginning result here, but there were many generalizations worked out in the 19th century, culminating in the development of class field theory in the first half of the 20th century. The Langlands program is in part about the development of non-abelian reciprocity laws.
Behaviour of arithmetic functions: A typical question here would be to investigate behaviour of functions such as $d(n)$ (the function which counts the number of divisors of a natural number $n$). These functions often behave quite irregularly, but one can study their asymptotic behaviour, or the behaviour on average.
Diophantine approximation and transcendence theory: The goal of this area is to establish results about whether certain numbers are irrational or transcendental, and also to investigate how well various irrational numbers can be approximated by rational numbers. (This latter problem is the problem of Diophantine approximation). Some results are Liouville's construction of the first known transcendental number, transcendence results about $e$ and $\pi$, and Roth's theorem on Diophantine approximation (for which he got the Fields medal).
The theory of modular (or more generally automorphic) forms: This is an area which grew out of the development of the theory of elliptic functions by Jacobi, but which has always had a strong number-theoretic flavour. The modern theory is highly influenced by ideas of Langlands.
The theory of lattices and quadratic forms: The problem of studying quadratic forms goes back at least to the four-squares theorem of Lagrange, and binary quadratic forms were one of the central topics of Gauss's Disquitiones. In its modern form, it ranges from questions such as representing integers by quadratic forms, to studying lattices with good packing properties.
Algebraic number theory: This is concerned with studying properties and invariants of algebraic number fields (i.e. finite extensions of $\mathbb Q$) and their rings of integers.
There are more topics than just these; these are the ones that came to mind. Also, these topics are all interrelated in various ways. For example, the prime counting function is an example of one of the arithmetic functions mentioned in (4), and so (1) and (4) are related. As another example, $\zeta$-functions and $L$-functions are basic tools in the study of primes, and also in the study of Diophantine equations, reciprocity laws, and automorphic forms; this gives a common link between (1), (2), (3), and (6). As a third, a basic tool for studying quadratic forms is the associated theta-function; this relates (6) and (7). And reciprocity laws, Diophantine equations, and automorphic forms are all related, not just by their common use of $L$-functions, but by a deep web of conjectures (e.g. the BSD conjecture, and Langlands's conjectures). As yet another example, Diophantine approximation can be an important tool in studying and solving Diophantine equations; thus (2) and (5) are related. Finally, algebraic number theory was essentially invented by Kummer, building on old work of Gauss and Eisenstein, to study reciprocity laws, and also Fermat's Last Theorem. Thus there have always been, and continue to be, very strong relations between topics (2), (3), and (8).
A general rule in number theory, as in all of mathematics, is that it is very difficult to separate important results, techniques, and ideas neatly into distinct areas. For example, $\zeta$- and $L$-functions are analytic functions, but they are basic tools not only in traditional areas of analytic number theory such as (1), but also in areas thought of as being more algebraic, such as (2), (3), and (8). Although some of the areas mentioned above are more closely related to one another than others, they are all linked in various ways (as I have tried to indicate).
[Note: There are Wikipedia entries on many of the topics mentioned above, as well as quite a number of questions and answers on this site. I might add links at some point, but they are not too hard to find in any event.]
As far as I know there is no way of finding the discriminant without finding the full ring of integers in the process.
Indeed, once you know the ring of integers, finding the discriminant is a trivial piece of linear algebra; and conversely, if you know the discriminant in advance, that makes finding the ring of integers much easier (because once you've found enough integers to generate a subring with the right discriminant, you know you can stop). So determining the discriminant cannot be all that much easier than determining the integers.
As for how to find the integers, it is a very well-studied problem; if you're interested in such things, Henri Cohen's "A course in computational algebraic number theory" is a very good reference.
Best Answer
I. Yes if $r$ is positive. Not necessarily if $r$ is negative. If $a_1,\ldots,a_{n-1}$ are algebraic integers, then any root of $x^n + a_{n-1}x^{n-1}+\cdots + a_1x + a_0$ is also an algebraic integer. Writing $r=p/q$ with $p$ and $q$ (positive) integers, it is clear that $(z-1/z)^p$ is an algebraic integer, and $(z-1/z)^r$ is a root of $x^q - (z-1/z)^p$, hence an algebraic integer itself. If $r$ is negative, then the answer may be negative: e.g., $z=i$, $z-(1/z) = i-(-i) = 2i$, and taking $r=-1$ yields $(2i)^{-1} = -\frac{i}{2}$, which is not an algebraic integer.
II. As mr_e_man noted, if $a-(1/a)$ is an algebraic integer, then $a$ is a root of $x^2 - (a-1/a)x -1$ which is a monic polynomial with algebraic integer coefficients, hence $a$ is an algebraic integer.
In both cases, the key is
So that’s the theorem you want to prove.