CEU Electronic Theses and Dissertations, 2012
| Author | Lapkova, Kostadinka Stoeva |
|---|---|
| Title | Class Number Problems for Quadratic Fields |
| Summary | The current thesis deals with class number questions for quadratic number fields. The main focus of interest is a special type of real quadratic fields with Richaud–Degert discriminants d = (an)^2 +4a, which class number problem is similar to the one for imaginary quadratic fields. The thesis contains the solution of the class number one problem for the two-parameter family of real quadratic fields with square-free discriminant d = (an)^2 +4a for positive odd integers a and n, where n is divisible by 43.181.353. More precisely, it is shown that there are no such fields with class number one. This is the first unconditional result on class number problem for Richaud–Degert discriminants depending on two parameters, extending a vast literature on one-parameter cases. The applied method follows results of A. Biró for computing a special value of a certain zeta function for the real quadratic field, but uses also new ideas relating our problem to the class number of some imaginary quadratic fields. Further, the existence of infinitely many imaginary quadratic fields whose discriminant has exactly three distinct prime factors and whose class group has an element of a fixed large order is proven. The main tool used is solving an additive problem via the circle method. This result on divisibility of class numbers of imaginary quadratic fields is applied to generalize the first theorem: there is an infinite family of parameters q = p_1p_2p_3, where p_1, p_2, p_3 are distinct primes, and q=3(mod 4), with the following property. If d = (an)^2 + 4a is square-free for odd positive integers a and n, and q divides n, then the class number of quadratic fields with this discriminant is greater than one. The third main result is establishing an effective lower bound for the class number of the family of real quadratic fields Q(\sqrt{d}), where d = n^2 + 4 is a square-free positive integer with n = m(m^2 − 306) for some odd m, with some extra condition on d. This result can be regarded as a corollary of a theorem of Goldfeld and some calculations involving elliptic curves and local heights. The lower bound tending to infinity for a subfamily of the real quadratic fields with discriminant d = n^2+4 could be interesting having in mind that even the class number two problem for these discriminants is still an open problem. Finally, the thesis contains a chapter on a joint work in progress with A. Biró and K. Gyarmati, which tries to solve the class number one problem for the whole family d = (an)^2 + 4a. |
| Supervisor | Biró, András |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2012/lapkova_kostadinka.pdf |
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