Summary: Automorphisms of Finite Rings and
Applications to Complexity of Problems
Manindra Agrawal and Nitin Saxena
National University of Singapore
In mathematics, automorphisms of algebraic structures play an important role.
Automorphisms capture the symmetries inherent in the structures and many
important results have been proved by analyzing the automorphism group of
the structure. For example, Galois characterized degree five univariate polyno-
mials f over rationals whose roots can be expressed using radicals (using ad-
dition, subtraction, multiplication, division and taking roots) via the structure
of automorphism group of the splitting field of f. In computer science too, au-
tomorphisms have played a useful role in our understanding of the complexity
of many algebraic problems. From a computer science perspective, perhaps the
most important structure is that of finite rings. This is because a number of
algebraic problems efficiently reduce to questions about automorphisms and iso-
morphisms of finite rings. In this paper, we collect several examples of this from
the literature as well as providing some new and interesting connections.
As discussed in section 2, finite rings can be represented in several ways.