We present a short and rather self-contained introduction to the theory of finite-dimensional division algebras, setting out from the basic definitions and leading up to recent results and current directions of research. In Sections 2-3 we develop the general theory over an arbitrary ground field k, with emphasis on the trichotomy of fields imposed by the dimensions in which a division algebra exists, the groupoid structure of the level subcategories 𝒟ₙ(k), and the role played by the irreducible...

We find examples of polynomials $f\in D[t;\sigma ,\delta ]$ whose eigenring $ℰ\left(f\right)$ is a central simple algebra over the field $F=C\cap \mathrm{Fix}\left(\sigma \right)\cap \mathrm{Const}\left(\delta \right)$.

Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum a_k{\mathrm{e}}^{\langle {\lambda }^k,z\rangle }$ for which $\left(\right|{\lambda }^k{|/\mathrm{e})}^{|{\lambda }^k|}k!\left|a_k\right|$ is bounded. Then $F$ is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. $F$ is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to $F$ have also been established.

We explicitly construct a particular real form of the Lie algebra ${𝔢}_7$ in terms of symplectic matrices over the octonions, thus justifying the identifications ${𝔢}_7\cong \mathrm{𝔰𝔭}(6,𝕆)$ and, at the group level, $E_7\cong \text{Sp}(6,𝕆)$. Along the way, we provide a geometric description of the minimal representation of ${𝔢}_7$ in terms of rank 3 objects called cubies.