What topic/subject is this? We can take a value $j$ and say $j^m = -1$.  Now ordinarily, for the complexes, we could simply say that $j = e^{2 \pi i / m}$.  I'm wondering, though, if there is a way to create a number system using this $j$, kind of like we use $i$ with the complexes.  I'm also wondering if we could then define a $k$ such that $k^n = j^m$.
In general, is there an algebra or some topic that deals with this?  I guess that I'm trying to create a ring or a quotient ring.  I'm hoping that someone can tell me if there's a subject that studies algebras such as this in detail.  For instance, what kind of algebra is this?
 A: You might be looking at cyclotomic fields. You do not go outside the complex numbers with these constructions, because the complex numbers are algebraically closed - every polynomial equation like $x^m+1=0$ with real coefficients (or complex coefficients) has a root in the complex numbers. So adding roots of polynomial equations doesn't get you "out of the system".
There are constructions like Quaternions and Cayley Numbers (Octonions) which are constructed from the complex numbers "in a similar way" to the construction of the complex numbers from the reals. These are genuinely larger, but the commutativity of multiplication is lost, and there are many square roots for $-1$, not just two.
A: Your intuition that you would be working with quotients is right.
No matter what number system $R$ you choose (integers, rational numbers, real numbers, or any other ring or field) you can define $R[j]/(j^m+1)$ where $j$ is an indeterminate. In this quotient ring, $j$ has the property that $j^m\equiv -1$. This is just a quotient of a polynomial ring.
If you want another thing $k$ such that $k^n=j^m$, you could look at $R[j,k]/(k^n-j^m)$, where $j$ and $k$ are indeterminates. Again since $k^n-j^m\equiv 0$ modulo the ideal, that is tantamount to saying that $k^n\equiv j^m$ in this ring.
The things you generate the ideal to mod out are usually called "relations," and you can introduce any number and combination of relations this way. You shouldn't get carried away though: if you introduce too may relations you may make the quotient collapse to be the zero ring :)
