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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 $mathbb{F}_{q^n}$ 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
We establish an analogue of the classical Polya-Vinogradov inequality for $GL(2, F_p)$, where $p$ is a prime. In the process, we compute the `singular Gauss sums for $GL(2, F_p)$. As an application, we show that the collection of elements in $GL(2,Z)
Fix $a in mathbb{Z}$, $a otin {0,pm 1}$. A simple argument shows that for each $epsilon > 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{frac12-epsilon}$. It is an open problem to show t
Let $p=2n+1$ be an odd prime, and let $zeta_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $overline{mathbb{Q}_p}$ of $mathbb{Q}_p$. We let $ginmathbb{Z}_p[zeta_{p^2-1}]$ be a primitive root modulo $pmathbb{Z}_p[zeta_{p^2
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.