# 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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### Number of elements in $\mathrm{GL}(n,p)$ with maximal order

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$. I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
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### Counting the image of a map of varieties using the trace formula

Suppose $f: X\to Y$ is a finite map of varieties over a finite field $\mathbb F_q$. Is there an etale constructible $\mathbb Q_\ell$ sheaf $\mathscr F$ on $Y$ which counts the number of rational ...
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### Probability that random points are affine subspace

I asked this question in Math stack exchange, but I think it is more relevant here. Suppose that $\mathbb{F}_q$ is a finite field with $q$ elements. Let $U = \{u_1, \ldots,u_m\}$ be a set of $m$ ...
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### Number triangle

This question arose just out of curiosity. Note the triangle of 0-1's below, whose construction is as follows. Choose any number, say 53 as done here. The first line of the triangle is the binary ...
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### Affine projection of polynomials for a given set of points

(Not sure this question fits here, I will remove it in case it doesn't) Let $F_{\mathrm{ML}}[x_1, \ldots,x_n]$ denote the set of multilinear polynomials over a finite field $F=\mathbb{F}_q$ (i.e. a ...
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Let $K$ be a (possibly infinite) field of characteristic $p$, and $L$ be a finite field extension of $\mathbb{F}_p$, so that $L$ is finite and $L/\mathbb{F}_p$ is Galois. Suppose $K \otimes_{\mathbb{F}... 0answers 56 views ### zeros of a polynomial in three variables over finite field Consider a homogeneous symmetric polynomial$f(x,y,z)=(x+y+z)((xy)^{q-1}+(yz)^{q-1}+(zx)^{q-1})+x^{2q-1}+y^{2q-1}+z^{2q-1}$of degree$2q-1$, where$q$is an odd prime power. Under what condition ... 0answers 104 views ### For which (g,q) does there exist a supersingular curve? We say a curve over a finite field$\mathbb F_q$is supersingular if it's Frobenius eigenvalues (on$H^1(X,\mathbb Z_\ell)$) are of the form$q^{1/2}\alpha$for$\alpha$a root of unity. As far as I ... 1answer 194 views ### Reference / Survey for finite field analog number theory It is folklore that many number theoretic results on prime numbers have a simpler-to-prove finite field analog. For example, on the one hand, the proof of the Prime Number Theorem $$\#\{\text{prime ... 1answer 346 views ### Upper bound for an exponential sum involving characters of a finite field Let q = p^n be a prime power, \alpha\in\mathbb{F}_{q} a primitive element of the finite field \mathbb{F}_q and denote by \chi a non-trivial additive character of \mathbb{F}_{q} . Set \... 1answer 40 views ### describing embedding U_3(q)<O_6^-(q), q even Let q=2^k. I need to explicitly construct U_3(q) as a subgroup of G=GO_6^-(q). It is well-known that G\cong U_4(q), and as a subgroup of the latter one has U_3(q) fixing a non-isotropic ... 1answer 138 views ### Image of a polynomial function x^2+y^2-x+y-axy over \mathbb{F}_p Let p be an odd prime and h(x)=x^2+ax+1 be an irreducible polynomial over the field \mathbb{F}_p. I need to prove that the function$$\Psi: \mathbb{F}_p^2 \longrightarrow \mathbb{F}_p, \quad (x,... 0answers 113 views ### A Lefschetz style formula for the$\ell^\infty$torsion of an Abelian variety over a finite field Let$A/\mathbb F_q$be an abelian variety over a finite field. Define$A_\ell = A[\ell^\infty](\mathbb F_q)$, the$\ell^n$($n\geq 0)$torsion points defined over the base field. I can assume$\ell \...
Let $p,q$ be two prime numbers and let $p<q.$ Also let, $\mathbb{Z}_p$ and $\mathbb{Z}_q$ denote the fields formed by integers modulo $p$ and modulo $q$ respectively (with respect to the modulo $p$ ...