Narrow Results

Let $D$ be an integral domain that is algebraic over $\mathbb{Z}$. It is shown that each directed maximal partial order on $D$ is an Archimedean total order. Let $F$ be a subfield of $ℝ$ and $C=F(i)$ be the complex field over $F$. As a consequence of the above result, if $F$ is algebraic over $\mathbb{Q}$, then $C$ does not have a directed partial order making it a partially ordered ring. In particular, $C$ cannot be a lattice-ordered ring. The result is proved for certain partially ordered algebras of quaternions as well.

In this paper, we prove that the rings of quaternions and of octonions over an arbitrary real closed field are algebraically closed in the sense of Eilenberg and Niven. As a consequence, we infer that some reasonable algebraic closure conditions, including the one of Eilenberg and Niven, are equivalent on the class of centrally finite alternative division rings. Furthermore, we classify centrally finite alternative division rings satisfying such equivalent algebraic closure conditions: up to isomorphism, they are either the algebraically closed fields or the rings of quaternions over real closed fields or the rings of octonions over real closed fields.

We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions ℍ ≃ ℝ_0,2, or to the real Clifford algebra ℝ_0,3. In the quaternionic case, the approach by means of Lagrange polynomials is new, and gives a complete solution of the interpolation problem. In the case of ℝ_0,3, such a problem is dealt with here for the first time. Elements of the recent theory of slice regular functions are used. Leaving apart the classical cases ℝ_0,0 ≃ ℝ, ℝ_0,1 ≃ ℂ and the trivial case ℝ_1,0 ≃ ℝ⊕ℝ, the interpolation problem on Clifford algebras ℝ_p,q with (p,q) ≠ (0,2), (0,3) seems to have some intrinsic difficulties.

We consider the arithmetic background of integral representations of finite groups over $p$-adic and algebraic number rings. Some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite $p$-groups with the extra congruence conditions are constructed, and some applications are given. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed.

We introduce two groups of duplication processes that extend the well known Cayley–Dickson process. The first one allows to embed every $4$-dimensional (4D) real unital algebra ${𝒜}$ into an 8D real unital algebra denoted by $\mathrm{\text{FD}}({𝒜}).$ We also find the conditions on ${𝒜}$ under which $\mathrm{\text{FD}}({𝒜})$ is a division algebra. This covers the most classes of known $4$D real division algebras. The second process allows us to embed particular classes of $4$D RDAs into $8$D RDAs. Besides, both duplication processes give an infinite family of non-isomorphic $8$D real division algebras whose derivation algebras contain $\mathrm{\text{su}}(2)$.

We study realization fields and integrality of characters of finite subgroups of ${\mathrm{\text{GL}}}_n(C)$ and related lattices with a focus on the integrality of characters of finite groups $G$. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, order generated by the character values, the number of minimal realization splitting fields, and in particular, consider the conditions of realizability in the terms of Hilbert symbols and quaternion algebras and some orders generated by character values over the rings of rational and algebraic integers.