Related papers: Cyclic cubic number fields with harmonically balan…

Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary…

For any number field K with non-elementary 3-class group Cl(3,K) = C(3^e) x C(3), e >= 2, the punctured capitulation type kappa(K) of K in its unramified cyclic cubic extensions Li, 1 <= i <= 4, is an orbit under the action of S3 x S3. By…

Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(\zeta_3)$ with $d>1$ a cube-free integer and $\zeta_3$ a primitive third root of unity. Denote by $C_{k,3}^{(\sigma)}$ the $3$-group of ambiguous classes of the…

Cyclic number fields of odd prime degree are constructed as ray class fields over the rational number field. They are collected in multiplets sharing a common conductor and discriminant. The algorithms are implemented in Magma and applied…

Conditionally on the Generalized Riemann Hypothesis (GRH), we prove the following results: (1) a cyclic number field of degree $5$ is norm-Euclidean if and only if $\Delta=11^4,31^4,41^4$; (2) a cyclic number field of degree $7$ is…

Let $K$ be a cyclic cubic field and $\mathcal{O}_K$ be its ring of integers. In this note we prove that all cyclic cubic number fields with conductors in the interval $ [73, 11971]$ and with class number one are Euclidean.

The isomorphism type of the Galois group G of finite 3-class field towers of quadratic number fields with 3-class group of type (9,9) is determined by means of Artin patterns which contain information on the transfer of 3-classes to…

Astonishing new discoveries with quartets and octets of cyclic cubic fields sharing a common conductor are presented. Four kinds of graphs describing cubic residue conditions among the prime divisors of the conductor enforce elementary bi-…

By means of parametrized presentations of finite metabelian 3-groups, it is proved that the coclass cc(M) of the second 3-class group M=Gal(F_3^2(K)/K) of any algebraic number field K with elementary bicyclic 3-class group Cl_3(K)=(3,3) is…

For an infinite family of monogenic trinomials $P(X) = X^3\pm 3rbX-b$ in $\mathbb{Z}\lbrack X\rbrack$, arithmetical invariants of the cubic number field $L = \mathbb{Q}(\theta)$, generated by a zero $\theta$ of $P(X)$, and of its Galois…

Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $\zeta_3$ be a primitive cube root of unity. Then $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$ is a pure metacyclic field with group $\mathrm{Gal}(\mathrm{k}/\mathbb{Q})\simeq S_3$.…

Let $N=K(\sqrt[3]{D})$ be a cubic Kummer extension of the cyclotomic field $K=\mathbb{Q}(\zeta_3)$, containing a primitive cube root of unity $\zeta_3$, with cube free integer radicand $D>1$. Denote by $f$ the conductor of the abelian…

In this paper, we continue the study of unit reducible fields as introduced in \cite{LPL23} for the special case of cyclotomic fields. Specifically, we deduce that the cyclotomic fields of conductors $2,3,5,7,8,9,12,15$ are all unit…

Let \(\Gamma=\mathbb{Q}(\sqrt[3]{n})\) be a pure cubic field with normal closure \(k=\mathbb{Q}(\sqrt[3]{n},\zeta)\), where \(n>1\) denotes a cube free integer, and \(\zeta\) is a primitive cube root of unity. Suppose \(k\) possesses an…

Suppose that $K$ is a field of characteristic 0, $K_a$ is its algebraic closure, $p$ is a prime, $q=p^r$ is a power prime. Suppose that $f(x) \in K[x]$ is a polynomial of degree $n > 4$ without multiple roots. Let us consider the…

In Question 5.2 of [5], Hung and Tiep asked the following: If $\alpha$ is a sum of $k$ complex roots of unity and $\mathbb{Q}_{c(\alpha)}$ is the smallest cyclotomic field containing $\alpha$, is it true that…

It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a…

It is known on the Generalised Riemann Hypothesis that there are precisely $13$ cyclic cubic fields that are norm-Euclidean. Unconditionally, there is a gap between analytic estimates which hold for all sufficiently large conductors and…

A {sign-circuit cover} $\mathcal{F}$ of a signed graph $(G, \sigma)$ is a family of sign-circuits which covers all edges of $(G, \sigma)$. The shortest sign-circuit cover problem was initiated by M\'a$\check{\text{c}}$ajov\'a, Raspaud,…

We say that a Hamilton cycle $C=(x_1,\ldots,x_n)$ in a graph $G$ is $k$-symmetric, if the mapping $x_i\mapsto x_{i+n/k}$ for all $i=1,\ldots,n$, where indices are considered modulo $n$, is an automorphism of $G$. In other words, if we lay…