The Gauss Class-Number Problems

In Articles 303 and 304 of his 1801 Disquisitiones Arithmeticae [Gau86], Gauss put forward several conjectures that continue to occupy us to this day. Gauss stated his conjectures in the language of binary quadratic forms (of even discriminant only, a complication that was later dispensed with). Since Dedekind’s time, these conjectures have been phrased in the language of quadratic fields. This is how we will state the conjectures here, but we make some comments regarding the original versions also. Throughout this paper, k = Q( √ d) will be a quadratic field of discriminant d and h(k) or sometimes h(d) will be the class-number of k. In Article 303, Gauss conjectures that as k runs through the complex quadratic fields (i.e., d < 0), h(k) → ∞. He also surmises that for low class-numbers, his tables contain the complete list of fields with those class-numbers including all the one class per genus fields. This innocent addendum caused much heartache when in 1934 Heilbronn [Hei34] finally proved that k(d)→∞ as d→ −∞ ineffectively. Thus it remained at that time impossible to even give an algorithm that would provably terminate at a predetermined time with a complete list of the complex quadratic fields of class-number one (or any other fixed class-number). By the “class-number n problem for complex quadratic fields”, we mean the problem of presenting a complete list of all complex quadratic fields with class-number n. We will discuss complex quadratic fields and generalizations in Sections 3 – 5. For real quadratic fields (i. e., d > 0), Gauss surmises in Article 304 that there are infinitely many one class per genus real quadratic fields. By carrying over this surmise to prime discriminants, we get the common interpretation that Gauss conjectures there are infinitely many real quadratic fields with class-number one. We call this the “class-number one problem for real quadratic fields”. This is completely unproved and, to this day, it is not even known if there are infinitely many number fields (degree arbitrary) with class-number one (or even just bounded). We will discuss two approaches each to the one class per genus problem for complex quadratic fields and the class-number one problem for real quadratic fields. Admittedly, I don’t have much hope currently for the first approaches to each problem but I think the questions raised are interesting. On the other hand, I