In this paper, first we introduce a new notion of commuting condition that $\phi {\phi }_{1}A=A{\phi }_1\phi$ between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$. Suprisingly, real hypersurfaces of type $\left(A\right)$, that is, a tube over a totally geodesic $G_2\left({ℂ}^{m+1}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ satisfying the commuting condition. Finally we get a characterization of Type $\left(A\right)$ in terms of such commuting...

We begin by giving a criterion for a number field $K$ with 2-class group of rank 2 to have a metacyclic Hilbert 2-class field tower, and then we will determine all real quadratic number fields $\mathbb{Q}\left(\sqrt{d}\right)$ that have a metacyclic nonabelian Hilbert $2$-class field tower.

Given an algebraic number field $k$ and a finite group $\Gamma$, we write $ℛ\left(O_k\left[\Gamma \right]\right)$ for the subset of the locally free classgroup $\mathrm{Cl}\left(O_k\left[\Gamma \right]\right)$ consisting of the classes of rings of integers $O_N$ in tame Galois extensions $N/k$ with $\mathrm{Gal}(N/k)\cong \Gamma$. We determine $ℛ\left(O_k\left[\Gamma \right]\right)$, and show it is a subgroup of $\mathrm{Cl}\left(O_k\left[\Gamma \right]\right)$ by means of a description using a Stickelberger ideal and properties of some cyclic codes, when $k$ contains a root of unity of prime order $p$ and $\Gamma =V⋊C$, where $V$ is an elementary abelian group of order $p^r$ and $C$ is a cyclic group of order $m\>1$ acting faithfully on...

The classical Stern sequence was extended by K.B. Stolarsky and the first author to the Stern polynomials $a(n;x)$ defined by $a(0;x)=0$, $a(1;x)=1$, $a(2n;x)=a(n;x^2)$, and $a(2n+1;x)=xa(n;x^2)+a(n+1;x^2)$; these polynomials are Newman polynomials, i.e., they have only 0 and 1 as coefficients. In this paper we prove numerous reducibility and irreducibility properties of these polynomials, and we show that cyclotomic polynomials play an important role as factors. We also prove several related results, such as the fact that $a(n;x)$ can only have simple zeros, and we state a...