# Questions tagged [finite-fields]

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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### "multi-dimensional" cyclotomic number

Let $F_q$ be the finite field with $q$ elements with characteristic $p$ and with $g$ being a primitive root. Let $N$ be a divisor of $q-1$ and let $C_0$ be the subgroup of $F_q^*$ with index $N$. Then ...
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For all primes up to $p=89$ there exists a product $Q=\prod_{j=1}^d(x-a_j)$ involving $d\geq (p-1)/4$ distinct linear factors $x-a_j$ in $\mathbb F_p[x]$ such that $Q'$ has all its roots in $\mathbb ... 0answers 72 views ### Sum of binary quadratic forms over inputs of equal Hamming weight$\DeclareMathOperator{\field}{\mathbb{F}}$Let$n$be a positive integer, and let$q : \field_2^{n} \rightarrow \field_2$be a quadratic form, specified in coordinates as $$q(x)=\sum_{i =1}^n \alpha_i ... 0answers 135 views ### Vanishing product of polynomials over finite fields (x_1-x_2-x_3+x_4)(x_2-x_4-x_3+x_1)(x_3-x_1)(x_4-x_2)\equiv 0 over \mathbb F_3. Take polynomials p_1,\dots,p_n over variables x_1,\dots,x_n such that p_i does not depend on x_i and \Pi_{i=... 1answer 190 views ### For which \beta \in \mathbb{F}_{p^k}, \{1,\beta,\beta^2,\cdots,\beta^{k-1}\} form a basis of \mathbb{F}_{p^k}? [closed] Let p be a prime, and let \mathbb{F}_{p^k} be a finite field. For which \beta \in \mathbb{F}_{p^k}, \mathcal{B}_{\beta}:=\{1,\beta,\beta^2,\cdots,\beta^{k-1}\} form a \mathbb{F}_p-basis of \... 1answer 235 views ### How large can the dimension of a 'Span of powers of a finite field basis' be? Let p be a prime. For finite field \mathbb{F}_{p^k} and d\in\mathbb{Z}^+, I am considering the following quantity, where we interpret the field \mathbb{F}_{p^k} also as a \mathbb{F}_p-vector ... 2answers 389 views ### Supersingular curves over \mathbb{F}_q and the splitting of p I'm looking at chapter 4 of Waterhouse's "Abelian varieties over finite fields"; and Theorems 4.1 and 4.2 seem to use the following fact: Suppose that E/\mathbb{F}_q is an elliptic curve ... 2answers 200 views ### Number of involutions in finite reductive groups Let G be a connected split reductive group over \mathbb{Z}. Let n be a positive integer. Let i_n(q) be the number of elements of G(\mathbb{F}_q) satisfying x^n=1. Question: Is there a &... 1answer 120 views ### Primes of the form p=u^2+1 and number of points on the elliptic curve x^3+a x z^2=y^2 z Let p be prime of the form p=u^2+1. For a \in \mathbb{F}_p,a \ne 0, define E_a : x^3+a x z^2=y^2 z Let B= \lfloor 2 \sqrt{p}\rfloor Must we have (\#E_a(\mathbb{F}_p) -p - 1) \in \{2,-2,B,-B\... 1answer 195 views ### Bounds for the number of points on projective hyperelliptic curves over finite fields Let C be projective hyperelliptic curve over finite field K. What are bounds for the number of points \#C(K)? The Hasse-Weil bound requires smooth curves, and hyperelliptic curves are not smooth ... 2answers 186 views ### Number of solutions of quadratic equation from a perfect pairing over \mathbb{Z}/p^n\mathbb{Z} Let p>2 be a prime number, V=\left(\mathbb{Z}/p^n\mathbb{Z}\right)^{2k+1}. The bilinear form$$B:V\times V \rightarrow \mathbb{Z}/p^n\mathbb{Z}$$is a perfect pairing. That is, mapping x\in V ... 0answers 140 views ### On hypergeometric functions over finite fields Let \mathbb{F}_q be a finite field of q elements. Let A,B,C,\cdots denote the multiplicative characters over \mathbb{F}_q, and let \overline{A} denote the inverse of A, i.e., A(x)\... 1answer 86 views ### \mathbb R and \mathbb F_2 rank in boolean matrix product By rank I imply rank over reals (\mathbb R). I consider two n\times n matrices A,B having entries in 0/1. The product below follows usual matrix product rules except xy is AND(x,y) and x+... 2answers 385 views ### Isomorphic endomorphism algebras implies isogenous (for abelian varieties over finite fields)? \newcommand{\F}{\mathbb{F}} \newcommand{\End}{\mathrm{End}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} I would like to know if the following is true: Proposition A : Let A_1, A_2 ... 0answers 105 views ### What can we say about the intersection of an algebraic and product set? This question is a bit vague by design. Let F be a field. I'm mostly interested in finite fields, but would also be interested in R or C. Let S \subset F^d be an algebraic set and let A = ... 1answer 109 views ### Curves sharing points over finite fields, and their mutual divisibility Consider in \mathbb{A}^2(\mathbb{F}_q) two \mathbb{F}_q-rational curves \mathcal{X}:f(x,y)=0 and \mathcal{Y}:g(x,y)=0, and let \mathcal{Y} be absolutely irreducible. Suppose also that \... 2answers 340 views ### A quantity associated to a field extension Let F\subset E be a field extension. So E has a natural structure of F-vector space. A vector subspace V\subset E is a special subspace if F\subset V and V is closed under the inverse ... 0answers 8 views ### On solutions to linear system amalgam Input: I. System of \Omega(t) linear polynomials in \mathbb F_2[x_1,\dots,x_{t}]. II. System of \Omega(t) linear polynomials in \mathbb Z[x_1,\dots,x_{t}]. Can we output a common 0/1 ... 1answer 170 views ### On a system of equations in \mathbb F_2 Input: System of \Omega(t) independent polynomials in \mathbb F_2[x_1,\dots,x_{t}] of degree O(t). Can we output a common solution of the system in polynomial time? Can we output parity of the ... 1answer 210 views ### p-adic logarithms with fixed precision Probably this is easy, but we would like to see it on paper. Let p be prime and D,g,n positive integers. Let A=g^n \bmod p^D. Let \log(p,a,D) be the p-adic logarithm with precision D. In ... 0answers 72 views ### Subspaces of vanishing permanent Suppose that p\ge 5 is a prime, n a positive integer divisible by p-1, and L<\mathbb F_p^n a subspace of dimension d=n/(p-1). Do there exist vectors l_1,\dotsc,l_n\in L such that the ... 1answer 257 views ### Lower bounds for class number of function fields with fixed q, growing g Let X be a smooth project curve of genus g over the finite field with q elements. Let h be \# \mathrm{Pic}^0(X)(\mathbb{F}_q). Weil showed that h \geq (\sqrt{q}-1)^{2g}. Lachaud and Martin-... 0answers 118 views ### Properties of pointless projective curves over finite fields? Probably not research level, feel free to downvote. We got construction of bounded degree projective curves with no points over finite fields. This construction generalizes to higher dimension. One of ... 0answers 288 views ### Why the curve x^2+y^2+y+1=0 has only one point over \mathbb{F}_{3^7}? According to both sagemath and Magma the curve x^2+y^2+y+1=0 has only one point over \mathbb{F}_{3^7}. The projective closure has only one point too. Q1 What hypothesis are missing to not violate ... 1answer 77 views ### Low-Hamming weight vectors in low-dimensional subspaces of \mathbb{F}_p^n What is the maximum number vectors of Hamming weight at most w in a d-dimensional subspace of \mathbb{F}_p^n, where w,d,p are constant and p is odd. (The Hamming weight is the number of ... 2answers 429 views ### Chevalley-Warning-Ax for double covers Let f(x_1,\ldots,x_n) be a polynomial of degree d with coefficients in the finite field \mathbb F_q and let V(f)\subseteq\mathbb F_q^n be its set of zeroes. Assume d<n. Then Chevalley ... 0answers 46 views ### On the real and finite field rank of a 0/1 matrix - II Let M\in\{-\ell,\dots,-1,0,+1,\dots,+\ell\}^{n\times n} be a matrix of rank r where \ell\geq1 such that there is a permutation matrix in \{0,1\}^{m\times m} of order 2\ell. Fix a permutation ... 1answer 134 views ### On the real and finite field rank of a 0/1 matrix - I Let M\in\{-1,0,+1\}^{n\times n} be a matrix of rank r. Consider the matrix f(M)\in\{0,+1\}^{mn\times mn} where 0 in M is replaced by m\times m all 0 matrix, +1 in M is replaced by m\... 1answer 272 views ### Least prime in Artin's primitive root conjecture Let a be an integer which is neither a square nor -1. Artin's conjecture states that there are infinitely many primes p for which a is a primitive root modulo p. My question is whether there ... 2answers 565 views ### On a matrix problem in the field \mathbb F_2 Given M a symmetric matrix in \mathbb F_2^{n\times n} having \mathsf{det}_\mathbb R(M)\neq0 (non-singular in reals) and satisfying PMP'=(M+J+I) or P(M+J+I)P'=M where P is a permutation ... 0answers 73 views ### Polynomial composition utilizing polynomials in two different finite fields At every n\in\mathbb N (all polynomials are of degree O(1)) is there g_{3,1}^{(n)},\dots,g_{3,k}^{(n)}\in\mathbb F_3[x_1,\dots,x_n] at k=\mathsf{poly}(n) and g_2^{(n)}\in\mathbb F_2[x_1,\dots,... 0answers 65 views ### What is the computational complexity of solving a highly underdetermined system? Let F be a finite field with q elements. Consider an underdetermined system of linear equations with m equations and n variables where n\gg m. What is the complexity of solving such a highly ... 0answers 186 views ### Finding (and saturating) a sharp Babenko-Beckner inequality for finite fields My question is a follow-up to Abdelmalek Abdesselam's recent post What makes Gaussian distributions special? Local field version? asking about various characterizations of (real-valued) Gaussian ... 0answers 121 views ### Distinguishing 0/1 unimodular or singular matrices having \mathsf{Permanent}\in\{0,1\}? Let \mathcal T_n=\{M\in\{0,1\}^{n\times n}:\mathsf{Per}(M)=\mathsf{Det}(M)\wedge\mathsf{Det}(M)\in\{0,1\}\} (restricted set unimodular or singular having permanent and determinant identical). \... 2answers 220 views ### How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there? How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there? As much as I have searched, I have not found any results that answer my question; not even for k = 1,2. 1answer 143 views ### Dominating sets in subtournaments of the Paley tournament For a tournament T, let \mathrm{dom}(T) be the order of a smallest dominating set in T. Let q be a prime power congruent to 3 mod 4 and let T_q be the Paley tournament on q vertices. Is ... 0answers 125 views ### Question on rank of matrices over \mathbb F_2 A is a square matrix in \mathbb F_2^{n\times n} of rank k\leq n-1. B is a square matrix in \mathbb F_2^{n\times n} of rank n. T is a square matrix in \mathbb F_2^{n\times n} of rank 1... 1answer 117 views ### Schur complement and depermuting an algorithm for \mathsf{determinant}\bmod2 Let$$M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$$be a matrix in \mathbb F_2^{n\times n} where A\in\mathbb F_2 and D\in\mathbb F_2^{(n-1)\times(n-1)} are square. Through the determinant ... 0answers 29 views ### On a (k,l)-monochromatic Hamming distance in \mathbb F_2? A (k,l)-monochromatic edit on a matrix M\in\mathbb F_2^{n\times n} is the operation$$M+A$$where$A\in\mathbb F_2^{n\times n}$is of rank$1$and number of$1$'s in$A$is$kl$and there are$l$... 1answer 138 views ### Polynomials vanishing on prescribed layers Given a prime$p$and an integer$n\ge p$, what is the smallest possible degree of a polynomial$Q\in\mathbb F_p[x_1,\dotsc, x_n]$such that$Q$vanishes on every vector$x\in\{0,1\}^n$of weight$w(x)...
The exact question I am interested in is the following. Fix a small $\varepsilon\in(0,1)$ and an integer $q\ge 2$ (you may assume that $q$ is prime if it helps though I believe it shouldn't matter too ...