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Register a new userNormal elements in finite fields
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Authors
Trevor Hyde
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We give a simple derivation of the formula for the number of normal elements in an extension of finite fields. Our proof is based on the fact that units in the Galois group ring of a field extension act simply transitively on normal elements.
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Let $mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) in mathbb{F}_q[x]$ with $textrm{deg}(f_i(x)) leq m_i$, for $i = 1, 2$, and such that the rational function $f_1(x)/f_2(x)$ belongs to a certain set which we define, we present a sufficient condition for the existence of a primitive element $alpha in mathbb{F}_{q^n}$, normal over $mathbb{F}_q$, such that $f_1(alpha)/f_2(alpha)$ is also primitive.
We give a lower bound on multiplicative orders of some elements in defined by Conway towers of finite fields of characteristic two and also formulate a condition under that these elements are primitive
By definition primitive and $2$-primitive elements of a finite field extension $mathbb{F}_{q^n}$ have order $q^n-1$ and $(q^n-1)/2$, respectively. We have already shown that, with minor reservations, there exists a primitive element and a $2$-primitive element $xi in mathbb{F}_{q^n}$ with prescribed trace in the ground field $mathbb{F}_q$. Here we amend our previous proofs of these results, firstly, by a reduction of these problems to extensions of prime degree $n$ and, secondly, by deriving an exact expression for the number of squares in $mathbb{F}_{q^n}$ whose trace has prescribed value in $mathbb{F}_q$. The latter corrects an error in the proof in the case of $2$-primitive elements. We also streamline the necessary computations.
Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $Gamma$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the commensurability class of the field $mathcal{F}$ given by the compositum of the splitting fields of characteristic polynomials of generic elements of $Gamma$ determines the group $G$ upto isogeny over the algebraic closure of $K$.
Given a finite endomorphism $varphi$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K(varphi^{-infty}(alpha)) : = bigcup_{n geq 1} K(varphi^{-n}(alpha))$ generated by the preimages of $alpha$ under all iterates of $varphi$. In particular when $varphi$ is post-critically finite, i.e., there exists a non-empty, Zariski-open $W subseteq X$ such that $varphi^{-1}(W) subseteq W$ and $varphi : W to X$ is etale, we prove that $K(varphi^{-infty}(alpha))$ is ramified over only finitely many primes of $K$. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire in the case $X = mathbb{A}^1$ and Cullinan-Hajir, Jones-Manes in the case $X = mathbb{P}^1$. Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for $X = mathbb{P}^1$. The proof relies on Faltings theorem and a local argument.
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