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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Who was the first to prove that the automorphism group of a finite field is cyclic and is generated by the Frobenius automorphism?

$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\...
Taras Banakh's user avatar
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5 votes
1 answer
151 views

Rational functions of order $3$

Let $p$ be a prime, let $\mathbb K$ a field with $\operatorname{char}(\mathbb K)=p$ and $\mathbb{K}(x)$ be the field of fractions of the polynomial ring $\mathbb{K}[x]$, i.e. $$\mathbb{K}(x)=\left\{\...
Mersn's user avatar
  • 51
-2 votes
0 answers
131 views

Elimination over $\mathbb F_p[x,y]$

Let $p$ be a prime. Consider the two independent modular equations: $$a_1x^2+b_1y^2+c_1xy\equiv d_1\bmod p$$ $$a_2x^2+b_2y^2+c_2xy\equiv d_2\bmod p$$ Is it possible to extract the common roots $(x,y)\...
Turbo's user avatar
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0 votes
0 answers
165 views

Degree 6 Galois extension over $\mathbb{Q} $

Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
Sky's user avatar
  • 913
0 votes
0 answers
73 views

Number of solutions $x$ of equation $a_1 b_1^x + \dotsb + a_n b_n^x=0$ over a finite field

Let $F$ be a finite field and let $a_1, b_1, \dotsc, a_n, b_n \in F$ be field elements. I am interested in the number of solutions $0\leq x \leq |F|-1$ such that \begin{equation}\label{e:1} a_1 b_1^x +...
Albert Garreta's user avatar
10 votes
2 answers
483 views

Isomorphic finite fields of a skew field

Let $D$ be a skew field and $F$ and $E$ be isomorphic finite subfields of $D$, is it true that $F=E$?
Alborz Azarang's user avatar
1 vote
1 answer
165 views

Matrices over a finite field: matrices for which some unipotent $U$ satisfies Trace$(ZU)=0$ for all $Z$ in the commutant

Let $p$ be an odd prime number, let $A\in M_p(\mathbb{F}_p)$ be a $p$-by-$p$ matrix with coefficients in $\mathbb{F}_p$, let $C(A)$ be the commutant of $A$, and let $N\in M_p(\mathbb{F}_p)$ be a ...
loup blanc's user avatar
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0 votes
1 answer
109 views

Loss of degree for polynomials

Let $q$ be a power of a prime $p$, $m$ be an integer greater than $0$. Does there exist polynomials $P_0,\cdots,P_m$ of $\mathbb F_q[T]$ not all in $\mathbb F_q$ such that there exist polynomials $Q_0,...
joaopa's user avatar
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0 votes
1 answer
180 views

Constructing a family of $3$-wise independence functions from $\mathbb{Z}_p^n \rightarrow \mathbb{Z}_p$

A family of function hash functions $\mathcal{H}:\{h:N\rightarrow M\}$ is call $k$-wise independent if whenenver $h$ is drawn uniformly from $\mathcal{H}$, let $x_1,\ldots,x_k$ be distinct elements of ...
some1fromhell's user avatar
3 votes
0 answers
160 views

Characters of finite fields - Reference request

Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
Grad Student's user avatar
8 votes
1 answer
584 views

A question on algebraic independence

Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...
Rishabh Kothary's user avatar
4 votes
0 answers
156 views

When is $q$ invertible mod $m$, mod its order mod $m$, mod its order mod its order mod $m$, ad infinitum?

Fix an [edit: positive] integer $q$. Let me say that an [edit: positive] integer $m$ is IK over $q$ if $q$ and $m$ are coprime and the (multiplicative) order of $q$ mod $m$ is IK over $q$. Note that ...
Theo Johnson-Freyd's user avatar
0 votes
0 answers
113 views

A question on a system of quadratic polynomials

Consider the following system of quadratic polynomials $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,....,x_n]$ : $f_1 (\bar{x}) = x_1 + x_n^2 + q_1$ $f_i(\bar{x}) = x_i + q_i$ for $i \in \{2,...,n-1 \}$ $...
Rishabh Kothary's user avatar
2 votes
1 answer
126 views

Are there a few input bits that randomize the output of an $\mathbb{F}_2$ polynomial?

Suppose $f:\mathbb{F}_2^n\rightarrow \mathbb{F}_2$ is a degree $d$ polynomial and $\epsilon>0$ is some real number. Does there necessarily exist a set $C\subset [n]$ of coordinates with the size of ...
dankane's user avatar
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3 votes
0 answers
96 views

A question on the averages of Kloosterman sums

Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is, For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound $$\sum_{...
hofnumber's user avatar
  • 553
2 votes
0 answers
114 views

Can K$_3$ of finite fields be related to Teichmüller cocycles?

This is sort of a blind shot, but... For a ring $R$, its third algebraic K-group is given by $\operatorname K_3(R)=H_3(\operatorname{St}(R))$. To simplify matters, let $R$ be a finite field $\mathbb ...
მამუკა ჯიბლაძე's user avatar
4 votes
1 answer
226 views

Frobenius and regular scheme

Let $X$ be a noetherian regular scheme over $\mathbb{F}_{p}$. Then by Kunz's theorem, the absolute Frobenius $F: X\rightarrow X$ is flat and integral. Can it be written as a projective limit of finite ...
prochet's user avatar
  • 3,432
2 votes
2 answers
415 views

Defining a sign of square roots in GF(p)

$\DeclareMathOperator\GF{GF}$Consider the following expression: $$ \sqrt{a_1} \pm \sqrt{a_2} \pm \dots \pm \sqrt{a_n} = 0, $$ where $a_1, \dots, a_n$ are positive integers. We want to find the number ...
Oleksandr  Kulkov's user avatar
0 votes
0 answers
109 views

Characterisation of even characteristic quadratic system

$\DeclareMathOperator\supp{supp}$Let $f_i \in \bar{\mathbb{F}}_2[x_1,..,x_5]$ for $1 \leq i \leq 5$ be such that $f_1(\bar{x}) = x_1 + x_5^2 + q_1$, $f_2(\bar{x}) = x_2 + x_1^2 + q_2$, $f_3(\bar{x}) = ...
Rishabh Kothary's user avatar
1 vote
1 answer
105 views

A question on classification of quadratic polynomials in even characteristic

$\DeclareMathOperator\supp{supp}$Let $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,...,x_n]$ such that $f_i = x_i + q_i$ for $1\leq i \leq n-1$ and $f_n = q_n$ where $q_1,...,q_n$ are homogenous quadratic ...
Rishabh Kothary's user avatar
2 votes
2 answers
299 views

A graphic representation of classical unitals on 28 points

I would like to understand the geometry of the classical unitals. They are block designs containing $q^3+1$ points and whose blocks have cardinality $q+1$, where $q$ is a prime power. For $q=2$ (if I ...
Taras Banakh's user avatar
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1 vote
1 answer
95 views

Existence of a symmetric matrix satisfying certain irreducible conditions

Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...
Sky's user avatar
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1 vote
0 answers
67 views

Bias of $a^k / q$ modulo $q$?

Let $q$ be a prime. Let $0< a < q$ be an integer so that it is primitive modulo $q$. Let $k$ be a random integer up to $q-1$. Consider $$a^k = b_k + q * c_k$$ as $k$ varies modulo $q^2$. So $b_k$...
mtheorylord's user avatar
0 votes
1 answer
184 views

Trying to solve for the remainder of $a^q$ modulo $q$

Let $q$ be a prime and $a$ be a number from $0$ to $q-1$ (not an equivalence class). The elements $a^q$ are exactly the elements of order $q-1$ modulo $q^2$. I'm trying to solve the equation: $$a+2*\...
mtheorylord's user avatar
1 vote
1 answer
169 views

Algebraic independence of polynomials when truncated imply algebraic independence of the entire polynomial?

Let $f_1,\ldots,f_m \in \mathbb{F}[x_1,\ldots,x_n]$ and suppose $\hat{f}_i = f_i$ $\bmod \langle x_1,\ldots,x_n\rangle^3$ (i.e. the linear and quadratic part of $f_i$). Then if $\hat{f}_1,\ldots,\hat{...
Rishabh Kothary's user avatar
4 votes
0 answers
171 views

Intrinsic maps between complex integers modulo $p$ and integers modulo $p+2$

$\DeclareMathOperator\GF{GF}$Let $p$ and $p+2$ be twin primes. Let's assume that $-1$ is not a quadratic residue modulo $p$ (and therefore is a Q.R. modulo $p+2$). Consider the complex numbers $a+bi$ ...
mtheorylord's user avatar
2 votes
0 answers
68 views

Is the discrete logarithm equivalent to solving polynomial discrete logarithms?

Suppose we can quickly solve the discrete logarithm modulo $p$. Let's say $2$ is a generator so we can quickly find $l$ for which $2^l =h$ for any given target $h$. An interesting observation is that ...
mtheorylord's user avatar
1 vote
1 answer
212 views

On the estimate for a double exponential sum

I encounter a hyper-Kloosterman sum which needs some help from the experts here: For any integers $q,s \in \mathbb{N}^+$(which may not be necessarily co-prime with each other), how to bound the sum: $$...
hofnumber's user avatar
  • 553
0 votes
0 answers
88 views

A question on the evaluations of certain three-dimensional hyper-Kloostermans

There is a basic question regrading the 3-dimensional hyper-Kloosterman sum which needs some help from the experts here: For any integers $q,h \in \mathbb{N}$, how to estimate the sum: $$\sideset{_{}^{...
hofnumber's user avatar
  • 553
1 vote
0 answers
84 views

Functions that take quadratic residues to non quadratic residues

Let $p$ be prime and $Q$ be the set of integers $x$ mod $p$ so that $x^2-1$ is a quadratic residue. Let $Q^c$ be the complement of $Q$. If we don't consider $x = 1$ then these two sets have the same ...
mtheorylord's user avatar
4 votes
0 answers
253 views

Cosine Modulo $p$?

Consider the integers modulo a prime $p$. I'm looking for a nice polynomial function that acts as a sort of "cosine" on the integers modulo $p$. Specifically, I'm looking for solutions to ...
mtheorylord's user avatar
4 votes
1 answer
328 views

GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials

This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$. Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
Martin Brandenburg's user avatar
6 votes
0 answers
176 views

Newton type method for finite fields?

I have a polynomial $p(x)$ in $\mathbb{Z}/q\mathbb{Z}$ that is easy to compute for any $x$ but has an absurdly large degree $d > 2^{256}$. I know for a fact that it has a zero and I would like to ...
mtheorylord's user avatar
2 votes
0 answers
134 views

What is the periodicity of $((a^n \text{ modulo } p) \text{ modulo } q)$

This feels like it should be elementary but it came up in my research and I was not able to solve it. We can ask this question for any $p$ and $q$ but,let $p$ and $q$ be primes for simplicity. The ...
mtheorylord's user avatar
1 vote
1 answer
118 views

Constant term of a power modulo a polynomial

I'm interested in the constant term of $$(x+k)^m \in F_p[x]$$ modulo a polynomial $q(x)$ over the field $F_p$. The polynomial $q(x)$ is relatively simple in practice, take $q(x) = x^6 -2x^3+3$, for an ...
mtheorylord's user avatar
5 votes
1 answer
337 views

Fermat cubic hypersurfaces over finite fields

Consider the Fermat cubic $$ X = \{x_0^3+\dots +x_n^3 = 0\}\subset\mathbb{P}^n_{\mathbb{F}_{q}} $$ over a finite field $\mathbb{F}_{q}$ with $q$ elements. If $q \equiv 2 \mod 3$ then the projection $\...
Puzzled's user avatar
  • 8,832
4 votes
1 answer
203 views

Points on affine hypersurface over finite field

I am interested in the hypersurface $X\subset\mathbb{A}^4_{\mathbb{F}_{5^n}}$ defined by $$ X = \{x^3 + 3xy^2 + z^3 + 3zw^2 + 1 = 0\} $$ over a finite field $\mathbb{F}_{5^n}$ with $5^n$ elements. Via ...
Puzzled's user avatar
  • 8,832
2 votes
1 answer
182 views

Chinese remainder theorem for composition

Let $f(x) \in F_p[x]$ and I know $f(x)$ modulo two polynomials $\phi_1(x)$ and $\phi_2(x)$. What sort of information about $f$ modulo the composition $\phi_1(\phi_2(x))$ can I recover? I'm looking ...
mtheorylord's user avatar
6 votes
2 answers
397 views

Good and bad reduction for twists of algebraic curves

Suppose we have two curves $C/\mathbb{Q}$ and $C'/\mathbb{Q}$ which are twists of each other i.e. they are isomorphic over a field extension $K/\mathbb{Q}$. Suppose that $C$ has good reduction at a ...
did's user avatar
  • 595
2 votes
1 answer
145 views

Automorphism of positive characteristic field

Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$. I am interested in ...
Sky's user avatar
  • 913
2 votes
0 answers
186 views

Factorization of the polynomial $x^k + x^{k-1} + x^{k-2} + \cdots + x + 1$ in $\mathbb{F}_2[x]$ [closed]

Is anything known about the factorization of the polynomial $x^k + x^{k-1} + x^{k-2} + \cdots + x + 1$ in $\mathbb{F}_2[x]$? When can it be factored, what are the irreducible factors, what are the ...
José's user avatar
  • 209
5 votes
1 answer
298 views

Galois action on algebraic K-theory of finite fields

This might be well-known to experts. I was just teaching a course where we went through some parts of Quillen's theorem computing the higher algebraic K-theory of finite fields. Denote by $\mathbb F_q$...
Andreas Thom's user avatar
  • 25.3k
0 votes
1 answer
79 views

Algorithm to find a number B with same modulus as A with prime P and specific binary positions set to zero

Given a prime $P$, an integer $A$ $(0\leq A<P$), and a set of legal positions (encoded as a binary mask $\text{mask}$), is there an efficient algorithm to find a number $B$ that has the same ...
bang's user avatar
  • 3
1 vote
1 answer
277 views

Szemerédi–Trotter type theorem in finite field

This question is about the content of this paper by J. Bourgain, N. Katz, T. Tao. In the final step (page 18) of the proof of Szemerédi-Trotter type theorem, we have already known $$|A''+A''|\lesssim ...
Jian-An Wang's user avatar
2 votes
1 answer
145 views

On the estimate for the mixed 3-dimensional hyper-Kloosterman sum

There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum: For any positive integer $n$ not divisible by $p$, how to prove $$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p} \...
hofnumber's user avatar
  • 553
3 votes
0 answers
111 views

Effective Lang-Weil bounds for higher codimension varities

The Lang-Weil bounds imply that for a geometrically irreducible variety $X$ of dimension d over $\overline{\mathbb{F}_p}$ we have, $$|N_q(X)-q^{d}|\le O(q^{d-1/2}),$$ where $N_q(X)$ is the number of $\...
Niareh's user avatar
  • 145
4 votes
1 answer
329 views

Polynomial that is not always a square over $\mathbb{Z}_p$

Let $p > 3$ be prime. Is is true that there exists $x \in \mathbb{Z}_p$ such that $$ (1+x^2)^3-1 $$ is not a square in $\mathbb{Z}_p$? In particular, when $-1$ is not a square in $\mathbb{Z}_p$, ...
Dom's user avatar
  • 43
0 votes
0 answers
107 views

Cubic monic polynomial over z_p

Let $$ f_{a}(x)=x^3+(u-2-a)x^2+ax+1, $$ where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all $a\in\mathbb{Z}_p$ such that $f_{a}(x)$ factor linearly. Then what is the cardinality ...
user avatar
13 votes
2 answers
702 views

Isomorphism of varieties in $\mathbb{C}$ implies isomorphism over finite fields

Suppose that $X$ and $Y$ are algebraic varieties over $\mathbb{Z}$ (add your favourite hypotheses, like smooth or affine if needed). Denote by $X_k$ and $Y_k$ their base-change to varieties over a ...
a_g's user avatar
  • 445
1 vote
0 answers
63 views

Is there an available English translation for Artin's "Quadratische Körper im Gebiete der höheren Kongruenzen"?

Otherwise, is it reasonable to work through the German edition with only a basic knowledge of German?
delpsi's user avatar
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