15 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…

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 $L = \mathbb{Q}(\sqrt{A}, \sqrt{B}, \sqrt{C})$ 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 $Q_{1,1}$ 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 $p\ge 5$ be a prime and let $n\ge 3$. For $\alpha\in\mathbb{F}_{p^n}$, call $\alpha$ $k$-normal over $\mathbb{F}_p$ if the $\mathbb{F}_p$-vector space spanned by … has dimensio…

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…

Let $\mathbf{F}_{k}$ be the free group of rank $k$, and let $\widehat{\mathbf{F}}_{k}$ be its free profinite completion. An element is primitive if it belongs to a basis of the rel…

Let $\mathbf{F}_{k}$ be the free group of rank $k$. A nontrivial word $w\in\mathbf{F}_{k}$ is primitive if it belongs to some free basis. A finitely generated subgroup…