Let’s examine the role of topology in the study of fields and arithmetic. A topology on a field compatible with the field operations is given by an absolute value, which in turn defines a metric. Outside of number theory, people usually mean the standard real absolute value when they talk about an absolute value. Note that on this absolute value carries the Archimedian property that is an unbounded set. What about the other case?
We say an absolute value is non-Archimedian if instead of the standard triangle inequality we have the inequality (note that under this equality if is an integer, so the integers are a bounded set under their canonical image in our field). If you are not familiar with this, I don’t know of a better gentle introduction than Fernando Gouvea’s Introduction to the-adic numbers. Very briefly, we can define an absolute value on by defining it on . If is a prime and is a nonzero integer then we can write where are nonzero integers and does not divide . Then we can define to be the -adic absolute value on . So we have the real topology, the discrete topology and the -adic topologies on . The following important theorem says that this is it(at least for those which “respect the field operations” as we put it).
Theorem(Ostrowski’s Theorem): If is an absolute value on then the topology it defines is homeomorphic to that generated by the real absolute value, the discrete absolute value or one of the -adic absolute values.
This is a magnificent theorem which can be proved using very classical analytic methods . Certainly of interest is the topic of completions, which are well-covered in any undergraduate analysis course. With respect to any of these absolute values we can complete the rational numbers and get a new field which strictly contains the rationals and is often easier to work with. If we complete the rationals with respect to the discrete valuation… we just get the rationals again because any Cauchy sequence is already eventually constant. If we complete with respect to the real absolute value, we get and if we complete with respect to a -adic absolute value, we get .
There is another theorem of Ostrowski’s, this time on completions, which we note here:
Theorem( Ostrowski’s [Big] Theorem): A field complete with respect to an Archimedian absolute value is either or .
I use the terminology “Big” in parallel to those two Picard Theorems in Complex Analysis, and also because we use “Little” Ostrowski to prove this theorem. The key observation is that if has nonzero characteristic as a ring then the integers have finite image in , and thus form a bounded set. Hence has characteristic zero, so and thus inject into . Then the absolute value restricted to must still be archimedian and thus equivalent to the real absolute value, so injects into . Then some analysis can show that any element of is degree 2 algebraic over .
So what to do with non-Archimedian absolute values on a field ? Note that each defines a valuation ring made up of the elements of size less than or equal to one and a valuation ideal of elements of size strictly less than one. If the supremum of the absolute values of elements in the valuation ideal is not 1, we call a discrete valuation ring. The valuation ideal is maximal since everything in of size exactly 1 has an inverse in the valuation ring. The quotient is a field called the residue field and with this in mind we use the following (non-standard, but often used) terminology.
Definition: We call a local field if it is complete with respect to a non-Archimedian absolute value, its valuation ring is a discrete valuation ring and its residue field is finite.
Using this definition of a local field, we can give the following complete characterization:
Theorem: If is a local field, it is a finite extension of either or (the Laurent series field over ).
The reason local fields are so popular is the following:
Theorem(Hensel’s Lemma): If is a local field, its valuation ring, its valuation ideal and a polynomial over such that not all of its coefficients are in then if factors into relatively prime polynomials in the residue field then there exist lifts such that , .
Just like “Local Fields” there are many definitions of “Hensel’s Lemma” of varying strengths. For instance, I do not need to be a local field here, only complete with respect to a non-Archimedian absolute value. Possibly the strongest possible version is the case that if is a valuation ring(not necessarily discrete) and its fraction field is complete with respect to the given non-Archimedian absolute value and is a variety over , then a smooth rational point in the special fiber(over the residue field) gives way to a smooth point on the generic fiber (over ). For a proof, see Silverman’s Advanced Topics in the Arithmetic of Elliptic curves, Theorem IV.6.4.
Note also that he speaks in terms of Henselian rings and fields. That’s where we’re going next.