Additive Rank-1 Preservers Between Hermitian Matrix Spaces Over Quaternion Division Algebra

Let Q be the quaternion division algebra over real field F, Denote by Hn（Q） the set of all n x n hermitian matrices over Q. We characterize the additive maps from Hn（Q） into Hm（Q） that preserve rank-1 matrices when the rank of the image of In is equal to n. Let QR be the quaternion division algebra over the field of real number R. The additive maps from Hn （QR） into Hm （QR） that preserve rank-1 matrices are also given.

Cyclic Division Algebras with Non-norm Elements

In this paper, we construct some cyclic division algebras （K/F,σ,γ）. We obtain a necessary and sufficient condition of a non-norm elementγ provided that F = Q and K is a subfield of a cyclotomic field Q（ζpu）, where p is a prime and ζpu is a pu th primitive root of unity. As an application for space time block codes, we also construct cyclic division algebras （K/F,σ, γ）, where F = Q（i）, i = √-1, K is a subfield of Q（ζ4pu） or Q（ζ4pu1 pu2）, and γ = 1＋i. Moreover, we describe all cyclic division algebras （K/F, σ, γ） such that F = Q（i）, K is a subfield of L = Q（ζ4pu1, pu2） and γ= 1 ＋i, where [K： F] = φ（pu1 pu2）/d, d = 2 or 4, φ is the Euler totient function, and p1,p2 ≤ 100 are distinct odd primes.

On Discrete Hopf Fibrations, Grand Unification Groups, the Barnes-Wall, Leech Lattices, and Quasicrystals

A discrete Hopf fibration of S15 over S8 with S7 (unit octonions) as fibers leads to a 16D Polytope P16 with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Λ16. It is argued (conjectured) how a subsequent 2-1 mapping (projection) of P16 onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 241 polytope in 8-dimensions, and such that one can capture the chain sequence of polytopes 241,231,221,211in D=8,7,6,5dimensions, leading, respectively, to the sequence of Coxeter groups E8,E7,E6,SO(10)which are putative GUT group candidates. An embedding of the E8⊕E8and E8⊕E8⊕E8lattice into the Barnes-Wall Λ16 and Leech Λ24 lattices, respectively, is explicitly shown. From the 16D lattice E8⊕E8one can generate two separate families of Elser-Sloane 4D quasicrystals (QC’s) with H4 (icosahedral) symmetry via the “cut-and-project” method from 8D to 4D in each separate E8 lattice. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC’s Q×Qspanning an 8D space. Similarly, from the 24D lattice E8⊕E8⊕E8one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC’s) Q×Q×Qwith H4 symmetry and spanning a 12D space.

THE PROOF OF FERMAT'S LAST THEOREM

i) Instead of x￣n+ y￣n = z￣n ,we use as the general equation of Fermat's Last Theorem (FLT),where a and b are two arbitrary natural numbers .By means of binomial expansion ,(0.1) an be written as Because a￣r-(-b)￣r always contains a +b as its factor ,(0.2) can be written as where φ_r =[a￣r-(-b)￣r]/ (a+b ) are integers for r=1 . 2, 3. ...n (ii) Lets be a factor of a+b and let (a +b) = se. We can use x= sy to transform (0.3 ) to the following (0.4)(iii ) Dividing (0.4) by s￣2 we have On the left side of (0.5) there is a polynomial of y with integer coefficient and on the right side there is a constant cφ/s .If cφ/s is not an integer ,then we cannot find an integer y to satisfy (0.5), and then FLT is true for this case. If cφ_n/s is an integer ,we may change a and c such the cφ_n/s≠an integer .

On Crossed Product Algebras over Henselian Valued Fields

Let D be a tame central division algebra over a Henselian valued field E,D be the residue division algebra of D,E be the residue field of E,and n be a positive integer.We prove that M_(n)(D)has a strictly maximal subfield which is Galois(resp.,abelian)over E if and only if M_(n)(D)has a strictly maximal subfield K which is Galois(resp.,abelian)and tame over E withГ_(K)■Г_(D),whereГ_(K)andГ_(D)are the value groups of K and D,respectively.This partially generalizes the result proved by Hanke et al.in 2016 for the case n=1.

A Proof of Wedderburn＇s Theorem

In this note we first introduce the finite weak Herstein condition, which is a property that holds for every finite field. For a field F satisfying the finite weak Herstein condition, we then characterize whether all the finite-dimensional division F-algebras are commutative. This gives an alternate proof of Wedderburn＇s Theorem.

Recent development of Faith conjecture

Is a semiprimary right self-injective ring a quasi-Frobenius ring？ Almost half century has passed since Faith raised this problem. He first conjectured “No” in his book Algebra II. Ring Theory in 1976, but changing his mind, he conjectured “Yes” in his article “When self-injective rings are QF： a report on a problem” in 1990. In this paper, we describe recent studies of this problem based on authors works and raise related problems.