Asymptotic existence results for primitive completely normal elements in extensions of Galois fields.

A primitive completely normal element for the extension E / F of Galois fields is a generator of the multiplicative group of E, which simultaneously is normal in E / K for every intermediate field K of E / F. Letting $$F=\mathbb {F}_q$$ und $$E=\mathbb {F}_{q^n}$$ , we denote by $$P_n(q)$$ the number of primitive elements of E, and by $$PCN_n(q)$$ the number of primitive completely normal elements for E / F. We prove that $$PCN_n(q) >0$$ provided that $$\ln (2) q \ge (t(n)-1)\cdot (\ln (2)+n\ln (q))$$ , where $$t(n):=\sum _{d|n}d$$ denotes the sum of all positive divisors of n and $$\ln $$ is the natural logarithm. Based on this estimate, we are able to prove that, for every fixed n, the quotient $$PCN_n(q)/P_n(q)$$ converges to 1 as q goes to infinity. We also achieve effective versions of the first result: $$PCN_n(q)>0$$ whenever $$q\ge n^{7/2}$$ and $$n\ge 7$$ , or when $$q\ge n^3$$ and $$n\ge 37$$ . The paper is self-contained and the proofs rely only on elementary combinatorics and number theory, as well as some basic calculus. [ABSTRACT FROM AUTHOR]