Research Projects

Let K be a number field and ??K its ring of integers. The class group, ??K of K is defined to be the quotient of the group of fractional ideals of ??K by the group generated by all non-zero principal ideals of ??K. ??K is always a finite group and its size is known as class number. This group is a measure of the extent to which unique factorization fails in ??K. It develops the subject of class field theory, and it can be used to prove the Kronecker-Weber theorem. In short, this group is a rich object worth studying. The following questions naturally arise in the study of ??K. Question 1: What is the size of ??K? For fixed integers n,d2, do there exist infinitely many fields of degree d whose class number is divisible by n? Can one construct families of such fields with large class numbers? Question 2: What is the structure of ??K? Question 3: Do the above questions have quantitative answers, depending, say, on the size of the discriminant of K? For any integer n2, the Cohen-Lenstra heuristic predicts that a positive proportion of quadratic number fields (in fact, any number fields) have class numbers divisible by n. Proving this heuristic seems out of reach with the current state of knowledge. On the other hand, Question 1 has been studied by authors for imaginary quadratic fields. However, the same needs to be studied for real quadratic fields and other number fields. The answers to the above questions help to understand the structure of the class groups as well as to improve/develop some tools to study some old conjectures. One of such conjectures is Gauss' class number one problem, which states that there are infinitely many real quadratic fields with class number one. For an integer n2, we define the n-rank of a field K to be the maximal integer r such that (ℤ/nℤ)ris a subgroup of ??K, and we denote it by rkn(K). This rank is closely related to the answer to Question 2. The following conjecture is widely believed to be true. Conjecture 1: For any integers n, d2, rkn(K) is unbounded when K runs through the fields of degree [K:ℚ]=d. When n=d, and more generally when n|d, this conjecture follows from Class Field Theory. On the other hand, when (n,d)=1, there is not a single case where Conjecture 1 is known to hold. When constructing families of degree d fields K with a given lower bound on rkn(K), it is natural to count the number of such fields constructed, ordered by discriminants. This is the quantitative aspect of Conjecture 1. The above problems are closely related to the solvability of certain exponential Diophantine equations. One of such class of equations is Lebesgue-Ramanujan-Nagell type equations, cx2+dm=yn, where c,d and are fixed positive integers. In a work, Prof. Michael Jacobson and Prof. Renate Scheidler mentioned that having a simple construction of small degree fields with large class groups might be useful in cryptography. Thus it would be interesting to construct lower degree fields with large class groups.