Division algebras and maximal orders for given invariants

Abstract

Brauer classes of a global field can be represented by cyclic algebras. Effective constructions of such algebras and a maximal order therein are given for Fq(t), excluding cases of wild ramification. As part of the construction, we also obtain a new description of subfields of cyclotomic function fields.

Primitive idempotents in central simple algebras over $\mathbb{F}_q(t)$ with an application to coding theory

“…We choose a finite place g (i.e., a monic irreducible polynomial) which is different from f . Using the algorithm from [1] we construct a division algebra D with Hasse invariants n−k n at f and k n at g (this splits at infinity since the sum of the Hasse-invariants is an integer). Using Lemma 14, we compute the local index of the central simple algebra A ⊗ D at f .…”

“…By Proposition 8, we can compute the set S of Hasse invariants of A. For the second step, construct a division algebra D (D should be given by an F q (t)-basis and structure constants), whose non-zero Hasse-invariants are exactly the elements of the set S. This can be done by the algorithm from [1]. We need to show that the denominator of each nonzero Hasse-invariant is relative prime to q.…”

“…Since the index of A is a divisor of n and (q, n) = 1, each of the s i is coprime to q. This implies that the algorithm from [1] can be applied. Note that this algorithm returns D in a cyclic algebra form, not in a structure constant from.…”

“…In [1] the authors propose a randomized polynomial time algorithm for constructing an F q (t)division algebra provided that the invariant at infinity is zero and the degree of the algebra is coprime to q. We propose an algorithm where the invariant at infinity is not necessarily zero when F q contains the nth roots of unity.…”

“…In Section 3 we provide randomized polynomial time algorithms for computing quaternion and symbol algebras with given invariants. In Section 4 we show how one can compute a division algebra D Brauer equivalent to a given central simple F q (t)-algebras, and an explicit isomorphism with the corresponding matrix ring over D, using either the algorithms from Section 3 or the algorithm from [1]. In Section 5 we construct constacyclic convolutional codes of designed Hamming distance and propose a polynomial time decoding algorithm.…”