Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quadratic Probabilistic Algorithms for Normal Bases   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 $\alpha \in K$
whose orbit $G\cdot\alpha$ forms an $F$-basis of $K$. Such an element $\alpha$
is called \emph{normal} and $G\cdot\alpha$ 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 $\alpha \in K$ is normal can be
reduced to deciding whether $\sum_{\sigma \in G} \sigma(\alpha)\sigma \in K[G]$
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 $G$.

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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
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The categories of this paper are: ao-sci