Class number in totally imaginary extensions of totally real function fields

We prove that almost real quadratic fields of a given type have a large ideal class number. For example, the number of ideal classes of the fields , where is the field of rational numbers, grows unbounded with , as ranges through all natural numbers, except for a very sparse sequence. An analogous fact is established for the fields of Ankeny-Brauer-Chowla, Amer. J. Math. 78 (1965), 51-61.

It was ascertained in previous chapters that upper bounds for the solutions of diophantine equations under our consideration depend essentially on the regulators of certain algebraic number fields related to the equation. Now we concentrate our attention on this phenomenon and relate it to the general problem of the magnitude of ideal class numbers. We show that algebraic number fields with ‘small’ regulator (hence ‘large’ class number) occur very frequently and in some sense constitute the majority of fields. Bounds for the solutions of the corresponding diophantine equations, e.g., Thue equations, are much better than the general bounds.

Ternary quadratic forms over number fields with small class number

Class Numbers and Iwasawa Invariants of Certain Totally Real Number Fields

Let {Km}m≥4 be the family of non-normal totally real cubic number fields as- sociated with the Q-irreducible cubic polynomials Pm(x) = x 3 − mx 2 − (m + 1)x − 1, m ≥ 4. We determine all these Km's with class numbers hm ≤ 3: there are 14 such Km's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents em of the ideal class groups of these Km go to infinity with m and we determine all these Km's with ideal class groups of exponents em ≤ 3: there are 6 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.

Article The Gauss-Hasse hypothesis on real quadratic fields with class number one was published on January 1, 1982 in the journal Journal für die reine und angewandte Mathematik (volume 1982, issue 335).

We compute an explicit spectral expansion and deduce an asymptotic formula for the fourth power moment of L-functions associated to a family of characters of a totally real number field with class number 1. In the second part we compute an upper bound for the second power moment Rankin-Selberg L-functions associated to newforms of special level over a totally real number field with class number 1.

In this paper, we consider a family of cubic fields $\{K_m\}_{m\geq4}$ associated to the irreducible cubic polynomials $P_m(x)=x^3-mx^2-(m+1)x-1,\,\,\,(m\geq4).$ We prove that there are infinitely many $\{K_m\}_{m\geq4}$'s whose class numbers are divisible by a given integer n. From this, we find that there are infinitely many non-normal totally real cubic fields with class number divisible by any given integer n.

We study the class number one problem for a one parameter family of real quadratic fields $\mathbb{Q}(\sqrt{d})$, where $d=100m^2+ 28m+2$ and $m$ is a positive integer. We show that there is no such field with class number one.

We determine all the simplest cubic fields whose ideal class groups have exponent dividing 3, thus generalizing the determination by G. Lettl of all the simplest cubic fields with class number 1 and the determination by D. Byeon of all all the simplest cubic fields with class number 3. We prove that there are 23 simplest cubic fields with ideal class groups of exponent 3 (and 8 simplest cubic fields with ideal class groups of exponent 1, i.e. with class number one).

Congruence function fields of genus g and class number g + 1 g + 1 are fully classified. As an application we determine explicitly the real quadratic function fields with this property and of odd characteristic for which the ring of integers is a unique factorization domain.

ABSTRACTA local ring is called a local ring of simple singularity of dimension one over the real field if it is isomorphic to a ring of the form ℝ{x,y}∕(f) and the number of proper ideals I of ℝ{x,y} with f∈I2 is finite. We first give a complete classification of local rings of simple singularity of dimension one over the real field. We also show that the rings have an infinitely generated prime cone unless they are isomorphic to either or , where n is a positive integer.

The generalized Rédei-matrix for function fields

In this paper the author describes the construction of a table of totally real cubic number fields. Each field is distinguished by the coefficients of a generating polynomial, the index of this polynomial over the field and the discriminant of the field. The class number and a fundamental pair of units is also given.

In this paper we consider the relationship between real quadratic fields, their class numbers and the continued fraction expansion of related ideals, as well as the prime-producing capacity of certain canonical quadratic polynomials. This continues and extends work in [10]–[31] and is related to work in [3]–[4].