Today I will be writing about something on Number Theory. Basically I just give an overview of the theory and knowledge from other fields of Mathematics that is required in the study (as much as I know of). Hence if you are not interested, do not read on anymore. I will not be responsible for any head injuries or damage caused while reading this article.
The study of number theory basically revolves over what we call integers. Since the objects we are studying has been identified, the next question will be how one determine whether two integers are equal. This is precisely the Fundamental Theorem of Arithmetic relating to the factorization of primes. The other important aspects is whether a given certain equation with integral cofficients has integral solution. One example of such equation is the Fermat's Last Theorem, where one asks if the equation x^{n} + y^{n} = z^{n} , where n is an integer greater or equal to 3, has integral solutions where none of x, y, z is zero. This statement looks simple but took Mathematicians about 300 years to finally to prove this (The final touch was by Andrew Wiles in around 1995). For the remainder of this article, I will briefly introduce to
you some parts of our "playground".
Algebraic Number Theory : This is the study of general number fields. A number field is a finite field extension over the rationals. Associated to each number field are algebraic invariants: the rings of integers, class group, unit group, residue field, discriminant etc. All these invariants give some form of arithmetic informations on the number fields. For example, the discriminant tells us about the so-called ramified primes. The rings of integers are examples of a Dedekind Domain which are rings having unique factorization of prime ideals. One of the standard approach to study a number field is to study its localization with respect to each prime ideals. This is useful since in its localization, we are only left with one prime to work with.
For instances in the base case, we have the p-adic integers and p-adic rationals. Also there is a topology on the localization given by a ultrametric, which gives a p-adic representation of the elements in the local field. Another tool that is also important in the study of number field is the Dedekind zeta function (which is a generalization of the zeta function). This function has a pole at 1 and its residue is related to class number and regulators of the number field (this is known as the class number formula).
Class Field Theory : The moral of this theory is that there is a relation between the arithmetic of a number field and its Galois extension. Another natural way (my personal feeling) to see this theory is as follow. The Gauss Quadratic Reciprocity Law (which one will learn in some elementary number theory course) in some sense determines the arithmetic of quadratic fields. Hence we are led to the question if there is similar of such law in a general number fields. The theory has been completely developed (between 1850 - 1925) for abelian number fields, i.e number fields which are Galois over the rational whose Galois group is abelian. The nonabelian case is still unsettled til today. In fact, the nonabelian theory has some relations to the so-called Langlands Program, which I will not dwell on it. There are two aspects of the theory, the Local Class Field Theory and the Global Class Field Theory. One final remark is that there is also generalization of the theory to other fields (function fields, higher local fields) other than number fields.
Homological Algebra : This evolved historically from Algebraic Topology where one is interested in studying the homology of certain (co)chain complexes. In the context of algebra, we are interested in the study of complexes in an abelian category (most important of all is the category of modules). In particular, if we restrict our extension to the category of modules over some group ring, we will obtain the so-called group cohomology theory. In number theory, we are interested in the case when the group is some Galois group of number fields or some other field of interest. We will say a bit on this in the next topic. Group Cohomology also provides an approach to the development of Class Field Theory. It also gives an cohomological interpretation of certain result in number theory (e.g Hilbert's Theorem 90).
Galois (Profinite) Cohomology : As we have remarked above, we are interested in the group cohomology when the group is a Galois group. Here we also deal with infinite Galois group and the group is not just an algebraic object, it also has a topology, given by the Krull topology or also known the profinite topology. Example of such groups are the Galois group of the algebraic closure of rationals (incidently the structure of this group is still unknown) or the Galois group of the maximal unramified extension outside a finite set of primes. These cohomology has a lot of arithmetic information encoded in it.
Algebraic K-Theory : This also evolved from Algebraic Topology in the study of vector bundles. In the case of rings, we have a list of invariant attached to a ring known as K_{i}(R). When the ring in consideration is a ring of integer, the zero K group is precisely the class group times a copy of the integer. The first K group is the unit group of the ring of integer. The second K group of a number field also has very interesting properties. For instance, it is isomorphic to the quotient of first Galois Cohomology with coefficient in (Q/Z)(2) by its maximal divisible sungroup (Frankly speaking, I have not fully understand this statement yet).
Unfortunately, my knowledge is still very limited. There is still areas like Iwasawa Theory, Elliptic Curves, Algebraic Geometry, Etale Cohomology et al which are currently just out of my grasp as yet. Also some of the topics I have above are not really fully exploited yet. Perhaps when I have learned more, I will come back to write a bit on these and expand some of the things I had written above. Finally thanks you for reading if you have read so far. I hope your brain is still in good shape...Don't worry too much, they are not examinable.
