Toru Nakahara

• Ph.D(Dr. of Science)
• University of MalakandSaga University

The aim of this paper is to determine that there does not exist any monogenic cyclic sextic field with prime power conductor except for the 7th, 9th cyclotomic fields and the maximal real subfield of conductor 13. Namely, we consider a problem of Hasse on the family of cyclic sextic fields, which is proposed by W. Narkiewitcz in general.

The purpose of this article is to determine the monogenity of families of certain biquadratic fields K and cyclic bicubic fields L obtained by composition of the quadratic field of conductor 5 and the simplest cubic fields over the field Q of rational numbers applying cubic Gauß sums. The monogenic biquartic fields K are constructed without using t...

The aim of this paper is to determine the monogenity of the family of cyclic sextic composite fields K · k over the field Q of rational numbers, where K is a cyclic cubic field of prime conductor p and k a quadratic field with the field discriminant d_k such that (p, d_k) = 1. Examples of our theorems are compared with the experiments by PARI/GP.

Let 𝑚 be a square free integer. The aim of this paper is to prove that the infinite family of pure octic field 𝐿 𝑄 √𝑚 is non-monogenic if 𝑚 𝑚𝑜𝑑 , ultimately, to complete the classification of pure octic fields 𝐿 𝑄 √𝑚 with respect to monogenity. We prove our results by considering the relative norms of the partial differents 𝜉 𝜉𝜎𝑗 of an integer 𝜉 fr...

Integral Basis and Relative Monogenity of Pure Octic Fields

An algebraic number ring is monogenic, or one-generated, if it has the form Z[α] for a single algebraic integer α. It is a problem of Hasse to characterize, whether an algebraic number ring is monogenic or not. In this note, we prove that if m is a square-free rational integer, m ≢ 1(mod 4) and m≢ ± 1(mod 9), then the pure sextic field L = Q(m6) is...

Let K be a composite field of a cyclotomic field kn of odd conductor n≧3 or even one ≧8 with 4|n and a totally real algebraic extension field F over the rationals Q and both fields kn and F are linearly disjoint over Q to each other. Then the purpose of this paper is to prove that such a relatively totally real extension field K over a cyclotomic f...

Let m ≠ 1 be a square-free integer. The aim of this paper is to construct an integral basis of the pure octic field L = Q(8 √ m) and to consider relative monogenity of L over its quartic subfield K = Q(4 √ m) as well as over its quadratic subfield k = Q(2 √ m). We prove that the field L is relatively monogenic over k for the case of m ≈ 5,13 (mod 1...

In this paper, we characterize whether the pure sextic fields ${\bf Q}(\sqrt[6]{m})$ with square-free integers m ≢ ±1(Mod 9) have power integral bases or do not; if m ≡ 2, 3 (Mod 4), then ${\bf Q}(\sqrt[6]{m})$ have power integral bases. We prove this by determining relative integral bases of such fields with respect to their cubic and quadratic su...

In this paper we characterize the monogenity of non-cyclic but abelian octic number fields over the rationals each of which is composed by a linearly disjoint cyclic quartic field of odd prime conductor and a quadratic field of prime discriminant In the case of each odd conductor the linear Diophantine equation with unit coefficients in a specified...

Let K be a real quadratic field Q(√n) with an integer n = df 2 with the field discriminant d of K and f ≥ 1. Q. Mushtaq found an interesting phenomena that any totally negative number k0 with k0 < 0 and k0σ < 0 belonging to the discriminant n, attains an ambiguous number km with K mkmσ < 0 after a finitely many actions k0Aj with 0 ≤ j ≤ m by modula...

Canonical Number System can be considered as natural generalization of radix representation of rational integers to algebraic integers. We determine the existence of Canonical Number System in two classes of pure algebraic number fields of degree 2n and n.

In this article we shall give a survey of Hasse’s problem for integral power bases of algebraic number fields during the last half of century. Specifically, we developed this problem for the abelian number fields and we shall show several substantial examples for our main theorem [7] [9], which will indicate the actual method to generalize for the...

Let K=ℚ(mn,mn,d 1 m 1 n 1 ℓ) be an octic field. If K is monogenic and a quadratic subfield ℚ(d 1 m 1 n 1 ) of K and a quartic subfield ℚ(mn,dn) are linearly disjoint, then K coincides with the field ℚ(-1,2,-3); namely K is equal to the cyclotomic field ℚ(ζ 24 ) [Y. Motoda and T. Nakahara, Arch. Math. 83, No. 4, 309–316 (2004; Zbl 1078.11061)]. In t...

Let K be a biquadratic field. M.-N. Gras and F. Tano gave a necessary and sufficient condition that K is monogenic by using a diophantine equation of degree 4 [3]. We consider algebraic extension fields of higher degree. Let F be a Galois extension field over the rationals \mathbbQ\mathbb{Q} whose Galois group is 2-elementary abelian. Then we shal...

In this paper we consider a subfield K in a cyclotomic field km of conductor m such that [km: K] = 2 in the cases of m = lpn with a prime p, where l = 4 or p > l = 3. Then the theme is to know whether the ring of integers in K has a power basis or does not.

Let K be the composite field of an imaginary quadratic field Q(ω) of conductor d and a real abelian field L of conductor f distinct from the rationals Q, where (d,f) = 1. Let ZK be the ring of integers in K. Then concerning to Hasse's problem we construct new families of infinitely many fields K with the non-monogenic phenomena (1), (2) which suppl...

The purpose of this paper is to exhibit a new family of real bicyclic biquadratic fields K for which we can write the Hasse unit index of the group generated by the units of the three quadratic subfields in the unit group E K of K. As a byproduct, one can explicitly relate the class number of K with the product of the class numbers of the three q...

Let m be a rational integer >1. We consider a generalized Fibonacci sequence {f j } modulo m defined by f j+2 =f j+1 +f j (j≥1) for arbitrarily given integers f 1 and f 2 . First, we classify the various periodic phenomena in them concerning a problem of divisors of each sequence. Second, we count the number of sequences {f j }, any term f j of whi...

Our aim is to give an arithmetical expression of the class number formula of real quadratic fields. Starting from the classical Dirichlet class number formula, our proof goes along arithmetical lines not depending on any analytical method such as an estimate for

The key feature at this conference was the 33 invited papers from the world's leading number theorists. Talks were in three sessions: analytical number theory; arithmetical algebraic geometry; and diophantive approximation. Speakers included: F.Beukers (University of Utrecht); R. Heath-Brown (Oxford); H.L. Montgomery (Ann Arbor, Michigan); T. Nakah...

In this note we shall prove that there exist infinitely many cyclic biquadratic fieldsK whose integral bases are neither {1, , 2, } nor {1, , , 3) for any numbers , inK. Next, we shall construct infinitely many cyclic biquadratic fieldsK which have the index 1, but still have not the integral basis {1, , 2, 3) for every inK. Finally we shall give a...

Let K be an abelian fleld whose Galois group is 2-elementary abelian over the rationals Q: If K is monogenic and it is generated by a qua- dratic subfleld and a quartic subfleld which are linearly disjoint, then K coin- cides with the fleld Q( p ¡1; p 2; p ¡3); namely K is equal to the cyclotomic fleld Q(‡24) (MN). In this article, we prove that al...

The authors show an arithmetical expression of Dirichlet’s class number formula for a real quadratic field of any field discriminant d. The proof of their arithmetical procedure depends on an estimate of the Dirichlet’s L-function L(s,χ) attached to the quadratic character χ with conductor d at s=1. Their theorem is a generalization of a result of...