A generalisation of Amitsur's A-polynomials
We find examples of polynomials $f\in D[t;σ,δ]$ whose eigenring $\mathcal{E}(f)$ is a central simple algebra over the field $F = C \cap {\rm Fix}(σ) \cap {\rm Const}(δ)$.
math.RA↗
arXiv subjects
Susanne Pumpluen
Publications and source records attributed to Susanne Pumpluen.
Irreducible skew polynomials over domains
Let $S$ be a domain and $R=S[t;σ,δ]$ a skew polynomial ring, where $σ$ is an injective endomorphism of $S$ and $δ$ a left $σ$ -derivation. We give criteria for skew polynomials $f\in R$ of degree less or equal to four to be irreducible. We apply them to low degree polynomials in quantized Weyl algebras and the quantum planes. We also consider $f(t)=t^m-a\in R$.
math.RA↗
The norm of a skew polynomial
Let $D$ be a finite-dimensional division algebra over its center and $R=D[t;σ,δ]$ a skew polynomial ring. Under certain assumptions on $δ$ and $σ$, the ring of central quotients $D(t;σ,δ) = \{f/g \,|\, f \in D[t;σ,δ], g \in C(D[t;σ,δ])\}$ of $D[t;σ,δ]$ is a central simple algebra with reduced norm $N$. We calculate the norm $N(f)$ for some skew polynomials $f\in R$ and investigate when and how the reducibility of $N(f)$ reflects the reducibility of $f$.
math.RA↗
How a nonassociative algebra reflects the properties of a skew polynomial
Let $S$ be a unital associative ring and $S[t;σ,δ]$ be a skew polynomial ring, where $σ$ is an injective endomorphism of $S$ and $δ$ a left $σ$-derivation. For each $f\in S[t;σ,δ]$ of degree $m>1$ with a unit as leading coefficient, we construct a unital nonassociative algebra whose behaviour reflects the properties of $f$. The algebras obtained yield canonical examples of right division algebras when $f$ is irreducible. We investigate the structure of these algebras. The structure of their right nucleus depends on the choice of $f$. In the classical literature, this nucleus appears as the eigenspace of $f$, and is used to investigate the irreducible factors of $f$. We give necessary and sufficient criteria for skew polynomials of low degree to be irreducible.
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Quotients of orders in algebras obtained from skew polynomials with applications to coding theory
We describe families of nonassociative finite unital rings that occur as quotients of natural nonassociative orders in generalized nonassociative cyclic division algebras over number fields. These natural orders have already been used to systematically construct fully diverse fast-decodable space-time block codes. We show how the quotients of natural orders can be employed for coset coding. Previous results by Oggier and Sethuraman involving quotients of orders in associative cyclic division algebras are obtained as special cases.
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Diagonal forms of higher degree over function fields of $p$-adic curves
We investigate diagonal forms of degree $d$ over the function field $F$ of a smooth projective $p$-adic curve: if a form is isotropic over the completion of $F$ with respect to each discrete valuation of $F$, then it is isotropic over certain fields $F_U$, $F_P$ and $F_p$. These fields appear naturally when applying the methodology of patching; $F$ is the inverse limit of the finite inverse system of fields $\{F_U,F_P,F_p\}$. Our observations complement some known bounds on the higher $u$-invariant of diagonal forms of degree $d$. We only consider diagonal forms of degree $d$ over fields of characteristic not dividing $d!$.
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Nonassociative cyclic extensions of fields and central simple algebras
We define nonassociative cyclic extensions of degree m of both fields and central simple algebras over fields. If a suitable field contains a primitive mth (resp., qth) root of unity, we show that suitable nonassociative generalized cyclic division algebras yield nonassociative cyclic extensions of degree m (resp., qs). Some of Amitsur's classical results on non-commutative associative cyclic extensions of both fields and central simple algebras are obtained as special cases.
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How to obtain lattices from (f,sigma,delta)-codes via a generalization of Construction A
We show how cyclic $(f,σ,δ)$-codes over finite rings canonically induce a $\mathbb{Z}$-lattice in $\mathbb{R}^N$ by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial $f$. This construction generalizes the one using certain $σ$-constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by $f$, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap coding. Previous results by Ducoat and Oggier are obtained as special cases.
cs.IT↗
Automorphisms and isomorphisms of Jha-Johnson semifields obtained from skew polynomial rings
We study the automorphisms of Jha-Johnson semifields obtained from an invariant irreducible twisted polynomial $f\in K[t;σ]$, where $K=\mathbb{F}_{q^n}$ is a finite field and $σ$ an automorphism of $K$ of order $n$, with a particualr emphasis on inner automorphisms and the automorphisms of Sandler and Hughes-Kleinfeld semifields. We include the automorphisms of some Knuth semifields (which do not arise from skew polynomial rings). Isomorphism between Jha-Johnson semifields are considered as well.
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Algebras whose right nucleus is a central simple algebra
We generalize Amitsur's construction of central simple algebras over a field $F$ which are split by field extensions possessing a derivation with field of constants $F$ to nonassociative algebras: for every central division algebra $D$ over a field $F$ of characteristic zero there exists an infinite-dimensional unital nonassociative algebra whose right nucleus is $D$ and whose left and middle nucleus are a field extension $K$ of $F$ splitting $D$, where $F$ is algebraically closed in $K$. We then give a short direct proof that every $p$-algebra of degree $m$, which has a purely inseparable splitting field $K$ of degree $m$ and exponent one, is a differential extension of $K$ and cyclic. We obtain finite-dimensional division algebras over a field $F$ of characteristic $p>0$ whose right nucleus is a division $p$-algebra.
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The automorphisms of Petit's algebras
Let $σ$ be an automorphism of a field $K$ with fixed field $F$. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras $K[t;σ]/fK[t;σ]$ obtained when the twisted polynomial $f\in K[t;σ]$ is invariant, and were first defined by Petit. We compute all their automorphisms if $σ$ commutes with all automorphisms in ${\rm Aut}_F(K)$ and $n\geq m-1$, where $n$ is the order of $σ$ and $m$ the degree of $f$,and obtain partial results for $n<m-1$. In the case where $K/F$ is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over $F$. We also briefly investigate when two such algebras are isomorphic.
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Solvable crossed product algebras revisited
For any central simple algebra over a field F which contains a maximal subfield M with non-trivial F-automorphism group G, G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit, and overlaps with a similar result by Albert which, however, is not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.
math.RA↗
Finite nonassociative algebras obtained from skew polynomials and possible applications to $(f,σ,δ)$-codes
Let $S$ be a unital ring, $S[t;σ,δ]$ a skew polynomial ring where $σ$ is an injective endomorphism and $δ$ a left $σ$-derivation, and suppose $f\in S[t;σ,δ]$ has degree $m$ and an invertible leading coefficient. Using right division by $f$ to define the multiplication, we obtain unital nonassociative algebras $S_f$ on the set of skew polynomials in $S[t;σ,δ]$ of degree less than $m$. We study the structure of these algebras. When $S$ is a Galois ring and $f$ base irreducible, these algebras yield families of finite unital nonassociative rings $A$, whose set of (left or right) zero divisors has the form $pA$ for some prime $p$. For reducible $f$, the $S_f$ can be employed both to design linear $(f,σ,δ)$-codes over unital rings and to study their behaviour.
cs.IT↗
Nonassociative differential extensions of characteristic p
Let F be a field of characteristic p. We define and investigate nonassociative differential extensions of F and of a central simple division algebra over F and give a criterium for these algebras to be division. As special cases, we obtain classical results for associative algebras by Amitsur and Jacobson. We construct families of nonassociative division algebras which can be viewed as generalizations of associative cyclic extensions of a purely inseparable field extension of exponent one or a central division algebra. Division algebras which are nonassociative cyclic extensions of a purely inseparable field extension of exponent one are particularly easy to obtain.
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Tensor products of nonassociative cyclic algebras
We study the tensor product of an associative and a nonassociative cyclic algebra. The condition for the tensor product to be a division algebra equals the classical one for the tensor product of two associative cyclic algebras by Albert or Jacobson, if the base field contains a suitable root of unity. Stronger conditions are obtained in special cases. Applications to space-time block coding are discussed.
math.RA↗
The nonassociative algebras used to build fast-decodable space-time block codes
Let $K/F$ and $K/L$ be two cyclic Galois field extensions and $D=(K/F,σ,c)$ a cyclic algebra. Given an invertible element $d\in D$, we present three families of unital nonassociative algebras over $L\cap F$ defined on the direct sum of $n$ copies of $D$. Two of these families appear either explicitly or implicitly in the designs of fast-decodable space-time block codes in papers by Srinath, Rajan, Markin, Oggier, and the authors. We present conditions for the algebras to be division and propose a construction for fully diverse fast decodable space-time block codes of rate-$m$ for $nm$ transmit and $m$ receive antennas. We present a DMT-optimal rate-3 code for 6 transmit and 3 receive antennas which is fast-decodable, with ML-decoding complexity at most $\mathcal{O}(M^{15})$.
cs.IT↗
A note on linear codes and nonassociative algebras obtained from skew-polynomial rings
Different approaches to construct linear codes using skew polynomials can be unified by using the nonassociative algebras built from skew-polynomial rings by Petit.
cs.IT↗
Space-time block codes from nonassociative division algebras
Associative division algebras are a rich source of fully diverse space-time block codes (STBCs). In this paper the systematic construction of fully diverse STBCs from nonassociative algebras is discussed. As examples, families of fully diverse $2\times 2$, $2\times 4$ multiblock and $4\x 4$ STBCs are designed, employing nonassociative quaternion division algebras.
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