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A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.

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During some digging of mine, I once found the following recursively defined family of polynomials: $P_0=P^2+2; P_{k+1}=P_k^2-2$. Using them one can show with purely algebraic means that the 2-adic ...
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Let $p ≥ 3$ be a prime and let $$E : y^2 = x^3 + A x + B$$ be an elliptic curve over $F_p$ with nonzero discriminant. I found the following pair of congruences for integers $ s, x, A, B, p:$ $$s^2 + ...
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In my work I commonly encounter large systems of polynomial equations for which it would be useful to know if there is a nontrivial solution over $\mathbb{F}_2$ (and if so, to find such a solution). ...
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$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, $\mathbb{F}_p$ be the finite field with $p$ elements and $\mathbb{F}_p^*=\mathbb{F}_p\setminus\{0\}$ be the multiplicative group of all ...
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I would like to know how much of the results on the Minimal Model Programme (MMP) over fields of finite characteristicb which are usually only stated for varieties over algebraically closed fields, ...
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We build a finite field $\mathbb Z/p\mathbb Z$, $p>2$. Then we introduce a sign function, which is $0$ at $0$, $+1$ at $1\dots(p-1)/2$, and $-1$ at $(p+1)/2\dots p-1$. Now we want to generalize the ...
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Let $E$ be supersingular elliptic curve defined over $\mathbb{F}_{p^2}$ and let $\ell$ be a prime. I'd like to know how one can check if E shares an edge, in the $\ell$-isogeny graph, with the set of ...
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Let $\mathcal{H} \subset \mathbb{P}^2(\mathbb{F}_{q^2})$ be the Hermitian curve, defined (up to projective equivalence) by $$ X^{q+1} + Y^{q+1} + Z^{q+1} = 0. $$ Its automorphism group is $\mathrm{PGU}...
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Let $F(X,Y)$ and $G(X,Y)$ be formal group laws over $\mathbf{F}_p$ of finite height. Hill proved (in this paper, Theorem E') that if $F$ and $G$ have the same characteristic polynomial of Frobenius, ...
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Question. Are there promising lines of attack to prove the following Conjecture? Conjecture. Suppose that $F$ is the finite field ${\rm GF}(p^n)$. Let $k\in\{1,\dots,n\}$ and set $d=\gcd(k,n)$. If $y\...
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Let $F$ be the finite field ${\rm GF}(2^n)$ of order $2^n$. Let $X=F\setminus\{0\}$ be its multiplicative group. Suppose that $n/d$ is odd where $k\in\{1,\dots,n\}$ and $d=\gcd(k,n)$, so that the ...
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The aim is for pairing inversion where miller inversion can only work if an equation is satisfied. So given a finite field modulus $q$ having degree $k$ ; and a finite field element $z$ having ...
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Let $p$ be a prime. I thought of ways to select a unique monic irreducible polynomial of degree n over a finite field with p elements. Most things failed, but this here seems to give at most 2 such ...
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Sorry if this question is too elementary for MO. Let $p$ be a prime and $F_p$ a field with $p$ elements and $F_{p^n}$ the field with $p^n$ elements Then we can choose an irreducible factor $f$ of ...
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Let $F$ be the finite field ${\rm GF}(p^n)$ of order $p^n$. The automorphism $\theta(x)=x^{p^k}$ of $F$ has order $n/d$ where $d=\gcd(n,k)$. Consider the function $f(x)=\theta(x)-x^{-1}$ from $X:=F\...
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Let $T_1 = |\{(U_1,U_2) \text{ $|$ } U_1 \text{ and } U_2 \text{ are affine flats of $\mathbb{F}_2^m$ such that } |dim(U_1) - dim(U_2)| \leq 1 \}|$ and T be the total number of affine flats of $\...
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Let $q=p^n$ be a prime power ($p$ is prime and $n\in\mathbb{Z}_{\ge 1}$) and $\mathbb{F}_q$ be a finite field with $q$ elements. Set $\widehat{\mathbb{F}_q^*}$ be the group of all multiplicative ...
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Recently, I encountered a double Kloosterman sum which needs some help from the experts here. My question is: Whether or not we can bound this double sum $$\hskip 0.5em \sideset{_{}^{}}{^{\ast}_{}}\...
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Let $A$ be an affine subspace of $\mathbb{F}_2^n$. Let $m\leq n$ and $Q_0, Q_1$ be linear maps $\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m$. Consider the following decision problem: Decide whether or not ...
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While experimenting with Newton's method for root finding $x \mapsto x-\frac{f(x)}{f'(x)}$ in positive characteristic, we observed that in the oddest characteristic $2$ there are always short cycles. ...
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Consider the space of $n \times n$ matrices over the finite field $GF(2)$. Is it possible to choose $k$ matrices $A_i, i = 1 ,\cdots k$, such that for every non-zero vector $b \in \{0,1\}^k$ the ...
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Let $n$ be a positive integer, and consider the hypersurface of singular $n\times n$ matrices over $\mathbb{F}_2$, denoted $$ \mathcal{S}_n = \{X\in M_n(\mathbb{F}_2) : \det(X)=0\}. $$ Note that \...
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Let $\Omega$ be the completion of $\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$ for the topology induced by valuation $-\deg$. One denotes by $|.|_\Omega$ the associated normalized norm ($|...
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Let $p$ be a prime number and denote $Q=(p^p-1)/(p-1)$. Is it true that $x^Q-x-1$ irreducible over $GF(p)[x]$?
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How to construct a full-rank $n \times m$ matrix over $\mathbb{F}_2$ with $m > n$, minimizing width and row sparsity? Goal: Construct a matrix $X \in \mathbb{F}_2^{n \times m}$, with $m > n$, ...
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$\;\;\;\;$ Fix a prime $p$, fix a power $q:=p^\aleph$, and consider the action of the Frobenius ring automorphism $\Psi:\vec{x}\mapsto\vec{x}^p$ on the product ring $\Bbb F_{q^m}^{(n)}:=\Bbb F_{q^m}\...
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I have a polynomial of degree around $2^{35}$ over a finite field $\mathbb{F}_p$, where $p$ is a 64-bit prime number. What is the most efficient algorithm for finding a root of such a polynomial? (...
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Let $X$ be an affine variety defined by homogeneous equations over $\mathbb{F}_q$ in $n + m$ variables. Let $W = X \cap \{ x_{n+1} = ... = x_{n + m} = 0\}$. I was wondering if there is a way to bound ...
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Consider a random $2 \times n$ matrix $X$ whose entries $X_{ij}$ take values in a finite field $E$ of characteristic $p$. Although the entries may not be independent, they satisfy the following non-...
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I stumbled upon an interesting mathematical pattern while working with polynomials over finite fields. While implementing an irreducibility test, I ended up using a weaker version of Rabin's test that ...
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Let $a,b,c,q$ be positive integers. One way to generalize the standard Kloosterman sum to two variables is $$ K(a,b,c;q) := \sum_{\substack{x_1\, \text{(mod $q$)}\\ (x_1,q)=1}} \sum_{\substack{x_2\, \...
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We know that finite fields have prime characteristic and we know a lot about them based in this fact. We can use that knowledge to establish very interesing and deep properties about these fields. In ...
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Let $\mathbb{F}_n$ denote the finite field with $n$ elements. Suppose that the (non-tall) matrix ${\bf M} \in \mathbb{F}_n^{r \times n}$, where $r \leq n$, has rank $r$ and for any $k \leq r$, the ...
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Reposting from MathStackExchange https://math.stackexchange.com/questions/5028883/determinant-of-an-endomorphism-of-a-drinfeld-module-over-a-finite-field with a slightly more general question. Let $\...
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We got a fast probabilistic algorithm for factorizations of univariate polynomials of special form modulo $N$ and would like to know if it is known or trivial. Let $N$ be composite integer with ...
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Given distinct lines $L_1,...,L_n \in \mathbb{F}_q^m$ and $\gamma \in (0,1)$, let $$f(L_1,..,L_n,\gamma) = |\{P \in \mathbb{F}_q^m \text{ $|$ } P \text{ lies in atleast }\gamma n \text{ lines} \}|$$ ...
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Let $G$ be a semisimple algebraic group over the finite field $\mathbf{F}_p$. Question 1. When is $G(\mathbf{F}_p)$ a solvable group? The obvious guess is "rarely". Due to some well-known ...
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Let $k$ be a field and $G$ a reductive $k$-group. The Whitehead group of $G$ is defined as the quotient $$W(k,G) := G(k)/G(k)^+$$ where $G(k)^+$ is the subgroup of $G(k)$ generated by the $k$-rational ...
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In this post, I don't require division algebras to be necessarily associative (but I do require a unit element). In synthetic projective geometry (part of incidence geometry), there is the notion of a ...
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Let $Q_1(x)=x^TA_1x$ and $Q_2(x)=x^TA_2x$ with $x\in GF(2)^n$, $A_i\in GF(2)^{n\times n}, i \in \{1, 2\}$. If $rank(A_1)=rank(A_2)=2$, is it possible that $Q_1(x)$ and $Q_2(x)$ can have the same roots ...
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If $Q(x) = x^T A x$ with $x \in GF(2)^n$ and $A \in GF(2)^{n \times n}$, is there a way to find how many roots $Q(x)$ has based on any properties of $A$ (e.g., rank, etc.)?
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Let $p \gg 1$ be a sufficiently large prime. I recently stumbled across a fascinating fact about the number of $\mathbb{F}_p$-points on elliptic curves over finite fields. Specifically, we have: Fact ...
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Let $V$ be an $n$-dimensional vector space over the finite field of cardinality $q$ and let $W_1,\ldots,W_m$ be hyperplanes of $V$ such that $$V=\bigcup_{i=1}^mW_i \,\,\hbox{ and }\,\,0=\bigcap_{i=...
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Let $f$ be a polynomial over a finite field $\mathbf{F}_p$ with $p \neq 2$. Let $R$ be the roots of $f$ in some extension field. I am interested in the multiset of differences $R - R = \{ r - s \mid r,...
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If $M$ is a full rank $n$ by $n$ binary matrix over $\mathbb{F}_2$, how much larger or smaller can the number of $1$s in $M^{-1}$ be, compared to the number of $1$s in $M$? Clearly the identity matrix ...
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Let $A$ be a uniformly random matrix in $\mathrm{M}_n(\mathbf{F}_p)$. It is well known that, as $p$ is fixed and $n$ tends to infinity, $A$ has repeated eigenvalues (over the algebraic closure $\...
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Let $p$ be a prime number and $q$ be a power of $p$. Fix a non-trivial additive character $\psi\colon \mathbb{F}_q\to \mathbb{C}^\times$ and, for each non-trivial multiplicative character $\chi\colon \...
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$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\PSL{PSL}$Consider the general linear group $\GL(n,q)$ over a finite field with $q$ elements and ...
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Let $n$ be a positive integer such that $n^2 + n + 1$ is a prime, and consider the Fermat surface $F$ given by the equation $X^n + Y^n - U^n - V^n = 0$ (where we work with homogeneous coordinates $(x :...
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The general linear group $GL_n(\mathbb{F}_2)$ acts on the powerset $2^{{\mathbb{F}_2}^n \setminus \{0\}}$ by multiplication: $A \cdot S := \{Ax \in {\mathbb{F}_2}^n : \, x \in S\}$, for an invertible ...
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