On the parity of the class number of multiquadratic number fields

This paper investigates the 2-class group of real multiquadratic number fields. Let p₁,p₂,…,pn be distinct primes and . We draw a list of all fields K whose 2-class group is trivial.     

Two-primary algebraic K-theory of two-regular number fields

We explicitly calculate all the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields, or maximal real subfields of such. Here 2-regular means that (2) does not split in the number field, and its narrow Picard group is of odd order. Received August 1, 1998     

On congruence relations between the fundamental units of biquadratic fields

Given any distinct prime numbers p,q, and r satisfying certain simple congruence conditions, we display a congruence relation between the fundamental units for the biquadratic field , modulo a certain prime ideal of OK. This congruence in particular implies the validity of the equivariant Tamagawa number conjecture formulated by Burns and Flach for the pair (h⁰(SpecK),Z[Gal(K/Q)]).     

On the number of solutions of a diophantine equation with symmetric entries

For a class of strictly increasing real valued functions f(n) we obtain an upper bound for the number of solutions of the equation

     

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

Exceptional units and Euclidean number fields

By a result of H.W. Lenstra, one can prove that a number field is Euclidean with the aid of exceptional units. We describe two methods computing exceptional sequences, i.e., sets of units such that the difference of any two of them is still a unit. The second method is based on a graph theory algorithm for the maximum clique problem. This yielded 42 new Euclidean number fields in degrees 8, 9, 10, 11 and 12. Received: 16 May 2006     

On some cubic or quartic algebraic units

Let ? be an algebraic unit such that rank of the unit group of the order Z[?] is equal to one. It is natural to ask whether ? is a fundamental unit of this order. To prove this result, we showed that it suffices to find explicit positive constants c₁, c₂ and c₃ such that for any such ? it holds that c₁c₂|?|?d??c₃|?|^2c₂, where d? denotes the absolute value of the discriminant of ?, i.e. of the discriminant of its minimal polynomial. We give a proof of this result, simpler than the original ones.     

Caliber number of real quadratic fields

We obtain lower bound of caliber number of real quadratic field using splitting primes in K. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if d is not 5 modulo 8. In both cases, we don't rely on the assumption on ζK(1/2).     

The algebraic structure of the set of solutions to the Thue equation

Let Fn be a binary form with integral coefficients of degree n?2, let d denote the greatest common divisor of all non-zero coefficients of Fn, and let h?2 be an integer. We prove that if d=1 then the Thue equation (T) Fn(x,y)=h has relatively few solutions: if A is a subset of the set T(Fn,h) of all solutions to (T), with r:=card(A)?n+1, then

(#)

h divides the numberΔ(A):=_∏1?k<l?rδ(ξk,ξl),

where ξk=〈xk,yk〉∈A, 1?k?r, and δ(ξk,ξl)=xkyl−xlyk. As a corollary we obtain that if h is a prime number then, under weak assumptions on Fn, there is a partition of T(Fn,h) into at most n subsets maximal with respect to condition (#).     

On the index of circular units in the full group of units of a compositum of quadratic fields

For a compositum of quadratic fields , where d₁,…,ds are square-free odd integers and , we study the group C of circular units of k. We construct a basis of C, compute the index of C in the full group of units of k and derive a lower bound for the divisibility of this index by a power of 2. These results give a lower bound for the divisibility of the class number of the maximal real subfield of k by a power of 2.     

On roots of polynomials with positive coefficients

We prove that an algebraic number α is a root of a polynomial with positive rational coefficients if and only if none of its conjugates is a nonnegative real number. This settles a recent conjecture of Kuba.     

Maximal class numbers of CM number fields

Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis and Artin's conjecture on the entirety of Artin L-functions, we derive an upper bound (in terms of the discriminant) on the class number of any CM number field with maximal real subfield F. This bound is a refinement of a bound established by Duke in 2001. Under the same hypotheses, we go on to prove that there exist infinitely many CM-extensions of F whose class numbers essentially meet this improved bound and whose Galois groups are as large as possible.     

On Fermat's equation in integral 2×2 matrices

Supported by the National Science Foundation of China     

On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Let m?−1 be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations

X³−mX²Y−(m+3)XY²−Y³=λ     

Diophantine equations over global function fields I: The Thue equation

We solve completely Thue equations in function fields over arbitrary finite fields. In the function field case such equations were formerly only solved over algebraically closed fields (of characteristic zero and positive characteristic). Our method can be applied to similar types of Diophantine equations, as well.     

The class number one problem for some totally complex quartic number fields

We prove that there are 95 non-isomorphic totally complex quartic fields whose rings of algebraic integers are generated by an algebraic unit and whose class numbers are equal to 1. Moreover, we prove Louboutin's Conjecture according to which a totally complex quartic unit εu generally generates the unit group of the quartic order Z[εu].