On definability of one-symbol languages in the monoid of finite languages with concatenation

Мы рассматриваем алгебру всех конечных языков над многосимвольным алфавитом с операцией конкатенации. Ранее было показано, что если взять подобную алгебру, но состоящую из всех регулярных многосимвольных языков, то в ней можно интерпретировать алгебру регулярных односимвольных языков, откуда следует, что теория обеих этих алгебр эквивалентна элементарной арифметике. В настоящей работе мы доказываем аналогичный результат для алгебры конечных языков: в ней определима подалгебра односимвольных языков, а сама она имеет теорию алгоритмически эквивалентную элементарной арифметике. We consider an algebra of all finite languages with the concatenation operation. For one-symbol languages it is known that its theory is equivalent to the first-order arithmetic. Earlier it was proved that for regular languages a one-symbol algebra can be interpreted in multi-symbol algebras. Here we show how to define a one-symbol subalgebra in multi-symbol algebras for finite languages.

ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC

In this article the $p$-essential dimension of generic symbols over fields of characteristic $p$ is studied. In particular, the $p$-essential dimension of the length $\ell$ generic $p$-symbol of degree $n+1$ is bounded below by $n+\ell$ when the base field is algebraically closed of characteristic $p$. The proof uses new techniques for working with residues in Milne–Kato $p$-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on $p$-symbol algebras (i.e, degree 2 symbols) result from this work. The generic $p$-symbol algebra of length $\ell$ is shown to have $p$-essential dimension equal to $\ell +1$ as a $p$-torsion Brauer class. The second is a lower bound of $\ell +1$ on the $p$-essential dimension of the functor $\operatorname{Alg}_{p^{\ell },p}$. Roughly speaking this says that you will need at least $\ell +1$ independent parameters to be able to specify any given algebra of degree $p^{\ell }$ and exponent $p$ over a field of characteristic $p$ and improves on the previously established lower bound of 3.

Relative Brauer group and pro-p Galois group of pre-p-henselian fields

Let p be a prime number and (F, v) a valued field. In this paper, we find a presentation for the p-torsion part of the Brauer group Br (F), by means of the valuation v. We only assume that F has a primitive pth root of the unity and the residue class field has characteristic not equal to p. This result naturally leads to consider valued fields that we call pre-p-henselian fields. It concerns valuations compatible with Rp, the p-radical of the field. To be precise, Rp is the radical of the skew-symmetric pairing which associates to a pair (a, b) the class of the symbol algebra (F; a, b) in Br F. In our main result, we state that pre-p-henselian fields are precisely the fields for which the Galois group of the maximal Galois p-extension admits a particular decomposition as a free pro-p product.

Some examples of division symbol algebras of degree 3 and 5

In this paper we provide an algorithm to compute the product between two elements in a symbol algebra of degree n and we find an octonion algebra (in general, without division) in a symbol algebra of degree three. Moreover, using MAGMA software, we will provide some examples of division symbol algebras of degree 3 and of degree 5.

Trace forms of Symbol Algebras

Let S be a symbol algebra. The trace form of S is computed and it is shown how this form can be used to determine whether S is a division algebra or not. In addition, the exterior powers of the trace form of S are computed.