Year: 2019
Type: preprint
Abstract: It is well known that for any finite Galois extension field $K/F$, with Galois group $G = \mathrm{Gal}(K/F)$, there exists an element $α\in K$ whose orbit $G\cdotα$ forms an $F$-basis of $K$. Such an element $α$ is called \emph{normal} and $G\cdotα$ is called a normal basis. In this paper we introduce a probabilistic algorithm for finding a normal element when $G$ is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether a random element $α\in K$ is normal can be reduced to deciding whether $\sum_{σ\in G} σ(α)σ\in K[G]$ is invertible. In an algebraic model, the cost of our algorithm is quadratic in the size of $G$ for metacyclic $G$ and slightly subquadratic for abelian $G$.
Source: arXiv (Cornell University)
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2921636962