27 problems

Let $\mathbb F_{q^n}/\mathbb F_q$ be a finite field extension. The extension possesses the MM property if it contains an element $a\in\mathbb F_{q^n}$ that is primitive and complet…

Let $q$ be a prime power, let $m$ be a positive integer, and identify $\mathbb{F}_{q^m}$ with an $m$-dimensional vector space over $\mathbb{F}_q$. Let $C=\{A_1,\ldots,A_m\}$ be a s…

Let $K$) be a number field of degree $d=[K:{\mathbb Q}]\geq 1$, with discriminant $\Delta_K$, and let $\theta\in K$ be primitive when $K={\mathbb Q}(\theta)$. Let $h$ denote the ab…

Let $q$ be a prime power and let $\mathbb{F}_q$ be a finite field. For each extension degree $n$, let $\mathbb{F}_{q^n}$ be the corresponding finite field, and let…

Conjecture. For every such $q$, $(q,7)\in A$.

Let $\mathbb F_q$ be a finite field with more than $2401$ elements. Cohen's conjecture. The field $\mathbb F_q$ contains four consecutive primitive elements. This conjecture concer…

Dartyge–Sárközy conjecture. If $p$ is large enough, then

Exceptional-pair conjecture. For the fields $\mathbb{F}_{q^m}$ with $m=3$ or $m=4$, the only possible exceptional pairs $(q,m)$ are

Let $t$ be a positive integer. A pair $(\alpha,\alpha^{-1})$ is a pair of primitive $1$-normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ when both elements are primitive a…

Let $k\in\{6,8,9,10,12\}$ and let … where $\deg f_1=\deg f_2=2$ and $f_2\nmid f_1$. A primitive pair $(\alpha,f(\alpha))$ in $\mathbb{F}_{2^k}$ is a pair whose two entries are prim…

Let $q=p^n$ be a prime power, let $\mathbb{F}_q$ be the finite field of size $q$, and call an element of $\mathbb{F}_q$ primitive if it generates $\mathbb{F}_q^*$. Cohen–Oliveira e…

Let $q=p^n$ be a prime power, let $\mathbb{F}_q$ be the finite field of size $q$, and call an element of $\mathbb{F}_q$ primitive if it generates $\mathbb{F}_q^*$. Define $q_0(4)$…

Sun's conjecture. If $p>13$ and $p\ne19,31$, then there is a primitive element $g$ of $\bb F_p$ such that $f(g)$ is a non-zero square in $\bb F_p$.

Primitive-root conjecture. The element $\theta$ is primitive in $\mathbb{F}_{p^p}$; equivalently, its multiplicative order is $p^p-1$.

Let $L = \mathbb{Q}(\sqrt{A}, \sqrt{B}, \sqrt{C})$ for distinct square-free non-zero integers $A$, $B$, and $C$, and let $M = \mathbb{Q}(\sqrt{A}, \sqrt{B})$. Here $\min\deg_L(v)$…

Let $q$ be a prime power. Write $\Gamma_2(2,2)$ for the set of positive integers $k$ such that, over $\mathbb{F}_{2^k}$, every relevant rational function represented by a matrix in…

Let $Q_{1,1}$ be the set of prime powers $q$ for which every rational function in $R_{1,1}$ admits a primitive pair $(\alpha,f(\alpha))$ with $\alpha\in\mathbb{F}_q$, and let…

Let $C_2$ be the least positive constant such that every prime power $q>C_2$ belongs to $Q_2$, where $Q_2$ is the set of prime powers $q$ for which every non-exceptional rational f…

Let $p$ be a prime, let $n=p$, let $a$ be a primitive element of $\mathbb{F}_p$, and let $\alpha\in\mathbb{F}_{p^p}$ be a root of the irreducible polynomial … Such an $\alpha$ is…

Let $A$ be the base ring and let $G\in\mathbb{F}_{q}\text{-}\operatorname{Gr}^{\boxplus}_{A}$. Suppose that there is a closed embedding … for some $N\in\mathbb{N}$. Let…

Let $A$ be the base ring, let $q$ be a power of the characteristic prime $p$, and let $G\in\mathbb{F}_{q}\text{-}\operatorname{Gr}^{+}_{A}$. Suppose that there is a closed embeddin…

Let $q$ be a prime power and let $\mathbb{F}_q$ denote the finite field with $q$ elements. Consecutive primitive elements are consecutive field elements that are all generators of…

Let $q$ be a prime power and let $\mathbb{F}_q$ denote the finite field with $q$ elements. Consecutive primitive elements are consecutive field elements that are all generators of…

Let $q$ be a prime power and let $\mathbb{F}_q$ denote the finite field with $q$ elements. A sequence of consecutive primitive elements is a sequence of consecutive elements of…

Let $q$ be a prime power and let $\mathbb{F}_q$ denote the finite field with $q$ elements. A field has consecutive primitive elements if some sequence of consecutive elements of it…