A lot of terms in abstract algebra – ramify, local, global – come from algebraic geometry, so it’s probably a good idea to explain what they’re all about. Algebraic number theory generally studies number fields and their rings of integers (as well as other objects – function fields, local fields, elliptic curves, L-functions – that help us understand number fields). Algebraic geometry typically studies rings like **C**[*x*1, *x*2, …, *x*(*n*)], the ring of polynomials in *n* variables over **C**, and their fraction fields.

For motivation of the term “local,” we might be interested in looking at polynomials in **C**[*x*] near *x* = 0. The ring **C**[*x*] has polynomials defined globally, i.e. for every *x*. Its fraction field, **C**(*x*), has quotients of polynomials, which many not be defined at some points. For instance, (*x* + 4)/(*x* – 3)(*x* – *i*) is not defined at 3 and *i*; But it’s defined near 0, e.g. for every point of distance to 0 of less than 1, so we should include it in an investigation of functions defined near 0.

The field **C** is algebraically closed, which means that every polynomial over **C** has a root. That means that the only irreducible elements of the PID **C**[*x*] are of the form *x* – *a* where *a* is some complex number. So every quotient in **C**(*x*) is a product of some linear polynomials divided by some other linear polynomials; and whenever *a* != 0, 1/(*x* – *a*) is defined near 0.

So if we’re just interested in quotients in **C**(*x*) that are defined near 0, we can just take all quotients without a factor *x* (i.e. *x* – *a*, *a* = 0) in the denominator. The set of all functions satisfying that condition forms a ring whose sole maximal ideal is the ideal of all functions that are 0 at 0, i.e. all functions that are divisible by *x*.

Generalizing this for rings in general, the a ring with only one maximal ideal is called a local ring, and to invert some elements of a ring – not necessarily in such a way that the result is a local ring – is to localize. If P is some prime ideal of a ring R, then the ring gotten from inverting all elements in R that are not in P is a local ring whose maximal ideal is P; this is called the local ring at P.

A global field is a field that in a certain sense contains global functions. Number fields are one example of a global field; the other is called function fields, which are finite extensions of **F**(*p*)(*x*), the field of functions in one variable over the finite field **F**(*p*). The term function field is algebraic-geometric, too; subject to a certain restriction, finite extensions of **F**(*p*)(*x*) are generated by a single element, with some minimal polynomial *f*(*y*). We can regard *f*(*y*) over the field** F**(*p*)(*x*) as *g*(*x*, *y*) over **F**(*p*) and then view the function field as the field of functions defined globally over the set of all elements (*x*, *y*) satisfying *g*(*x*, *y*) = 0.

A local field is trickier – the idea is to take a global field’s ring of integers (in the function field case, it’s an extension of **F**(*p*)[*x*]), extract a prime ideal out of it, localize with respect to the ideal, and then take something called a completion. If K is any field, then localizing K[*x*] at *x* = 0 and then completing yields K[[*x*]], the ring of all possibly infinite polynomials in *x*. That ring is local, since it can be shown that (*x*) is the only maximal ideal; its fraction field, denoted K((*x*)) will then be an example of a local field.