A Quick Algebraic-Geometric Note

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[x1, x2, …, 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)(xi) 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 xa 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/(xa) 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. xa, 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.

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