Skip to main content

Questions tagged [finite-fields]

Finite fields are fields (number systems with addition, subtraction, multiplication, and division) with only finitely many elements. They arise in abstract algebra, number theory, and cryptography. The order of a finite field is always a prime power, and for each prime power $q$ there is a single isomorphism type. It is usually denoted by $\mathbb{F}_q$ or $\operatorname{GF}(q)$.

Filter by
Sorted by
Tagged with
0 votes
0 answers
23 views

Find the number of matrices over the finite field $\mathbb F_{19}$, whose minimal polynomial has a certain degree $m$.

I am collaborating with some colleagues to create a TACA (a test assessing knowledge in Calculus, Linear Algebra, and Elementary Group Theory) practice test. During this process, I devised the ...
1048576's user avatar
  • 11
1 vote
0 answers
14 views

Orthogonal complement with respect to a subspace, and then with respect to the larger space.

Suppose I have the subspaces $W\leq V \leq \mathbb{F}_q^n$, with $n$ finite. Let $\langle ,\rangle\colon\mathbb{F}_q^n\times \mathbb{F}_q^n \rightarrow \mathbb{F}_q$ be the dot-product. If I then take ...
Johan's user avatar
  • 71
0 votes
1 answer
61 views

Finding BCH code syndromes

I' m not getting how syndromes are calculated for bch codes so I tried finding examples but still I don't seem to have it To calculate the first syndrome for the received message polynomial $R(x)=1+...
user159729's user avatar
1 vote
1 answer
60 views

How many roots are there of $(x^2-3)(x^3-3)$ in $K$, where $K$ is the splitting field of $x^3-1$ over $\mathbb F_{11}$?

Problem: How many roots are there of $(x^2-3)(x^3-3)$ in $K$, where $K$ is the splitting field of $x^3-1$ over $\mathbb F_{11}$? I checked that only $\bar 1 \in \mathbb F_{11}$ is root of $x^3-1 \...
Fuat Ray's user avatar
  • 1,140
4 votes
1 answer
162 views

Linear algebra question: does it have a solution?

Given $k\in\mathbb{N}$, $p$ a prime number, $s = (s_1, s_2,..., s_{2k+1})\in \mathbb{M}_{(2k+1)*1}(\mathbb{F}_p)$, the Hankel matrix generated by $s$ is denoted as $H$ where $$ H = \begin{pmatrix} s_1 ...
Youzhe Heng's user avatar
1 vote
0 answers
44 views

Singular locus of $\mathbb{F}_p[x,y,z]/(xy-z^2)$

How would I go about determining the singular locus of the hypersurface ring $R=\mathbb{F}_p[x,y,z]/(xy-z^2)$? I conjecture that the ring is regular at every maximal ideal except $(x,y,z)$. The ...
Anon's user avatar
  • 598
1 vote
1 answer
30 views

Equivalent polynomials over a finite field

Disclamer. I'm not good at math, and the last time I did it in school was 10 years ago, so I'm writing everything in my own words. Suppose we are working with polynomials in the space of remainders ...
ddvamp's user avatar
  • 13
1 vote
0 answers
53 views

Does there exists something like the BKK Theorem for polynomials over finite fields?

I'm trying to count the number of common $\mathbb{F}_q^\times$-zeros of some polynomials $f,g \in \mathbb{F}_q[x,y]$. I thought I had a solution using the BKK theorem but I reread the theorem ...
Amelia Gibbs's user avatar
0 votes
1 answer
90 views

Extended euclidian algorithm

I'm trying to understand how the matrix form of the extended euclidian algorithm for polynomials works for a BCH code with coefficients from $GF(2^4)$ in https://en.wikipedia.org/wiki/BCH_code for ...
user159729's user avatar
2 votes
0 answers
61 views

A conjecture on diagonal Ramsey numbers

Let $R(n,n)$ denote the $n$-th diagonal Ramsey number, i. e. the smallest integer $m$ such that any $m$-vertex graph contains either an $n$-clique or an $n$-independent set. Let us define a maximal $n$...
Bertrand Haskell's user avatar
6 votes
1 answer
75 views

Schur’s lemma over $\mathbb{F}_p$

I’m studying modular representation theory, and I got really stuck with the seemingly innocent statement. Consider $\mathrm{GL}_{2}(\mathbb{F}_{p})$ and its center $Z$, which is just a set of all ...
Matthew Willow's user avatar
5 votes
0 answers
68 views

Counting f-invariant subspaces over finite fields

Riffing off of On the number of $f$-stable subsets which I very much enjoyed thinking about, let $f : V \to V$ be a linear map on an $n$-dimensional vector space over a finite field $\mathbb{F}_q$ and ...
Qiaochu Yuan's user avatar
2 votes
0 answers
52 views

Finding Jordan Normal Form of ridiculous matrix.

As an exercise for our exams we were tasked with finding the jordan normal form of the following matrix where $\mathbb F_7$ denotes the finite field (modular arithemtic) over $\left\{0,1,2,3,4,5,6\...
Nils Schwebel's user avatar
1 vote
1 answer
27 views

Function from $ℕ$ to a non-prime finite field, compatible with addition, and invertible for small values

Let $n$ be some positive integer and $𝔽_q$ a finite field. I am looking for (sufficient and/or necessary) conditions on $𝔽_q$ for the existence of a map $φ:ℕ→𝔽_q$ that satisfies the following ...
Bruno's user avatar
  • 329
2 votes
2 answers
108 views

Irreducible factors of $X^n -1$ in $\mathbb{F}_q[X]$

I am stuck trying to prove the following corollary: Let $f = X^n -1 \in \mathbb{F}_q[X]$ with $gcd(q,n)=1$. Let k = order of q mod n. Then the degree of every irreducible factor of $f$ divides k. It ...
Very Interesting's user avatar
4 votes
0 answers
34 views

Minimum subset of the Grassmanian that covers all of a vector space $\mathbb{F}_d^n$

Consider a finite field $\mathbb{F}_d$ of order $d$, and let the vector space $V=\mathbb{F}_d^n$. Let $\mathbf{Gr}(m,V)$ be the Grassmanian containing all subspaces in $V$ of dimension $m$. Suppose $S\...
Damalone's user avatar
  • 329
0 votes
1 answer
79 views

Splitting Field of $x^4 - 10$ in $\mathbb{F}_7$

I am currently studying field theory and came across a problem that I need some help with. Specifically, I am interested in finding the splitting field of the polynomial $x^4 - 10$ over the finite ...
Khaled Alekasir's user avatar
5 votes
0 answers
93 views

Generic bound on quadratic character sum

Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free polynomial over $\mathbb{F}_q[x]$. Then by the Weil bound, we have the generic estimate $|\sum_{x\...
Madarb's user avatar
  • 623
2 votes
1 answer
59 views

Splitting field of $x^6 + 1$ over $F_2$

So I want to find the splitting field of $g(x)=x^6+1$ over $F_2$ and the degree of the extension, so what I have done is the following $$g(x)=x^6+1=(x^3)^2+1^2=(x^3+1)^2=(x+1)^2(x^2+x+1)^2$$ So we see ...
Donlans Donlans's user avatar
1 vote
0 answers
44 views

Conway Polynomial for p=2, n=3?

Im doing an exercise on Conway polynomials. As far as im concerned, for p=2, n=3 both $f(x)=x^3 + x^2 + 1$ and $g(x)=x^3 + x + 1$ satisfy every condition. According to every source i found, the latter ...
Vanessa K's user avatar
1 vote
1 answer
146 views

Simplest unsolvable quintic with one real root

I am aware that $t^5-t-1$ is unsolvable, but the proof I have seen involves a theorem linking its Galois group with the Galois group of its reduction mod $p$. If I wish to have a simpler proof (that ...
user21820's user avatar
  • 59.1k
2 votes
1 answer
59 views

Is there a finite dimensional vector space over a finite field with exactly two bases?

Is there a finite dimensional vector space over a finite field with exactly two bases? I searched and found that the answer is NO. But I have an example that $\mathbb{Z}_{3}$ is a 1-dimensional vector ...
Rattan verma's user avatar
3 votes
2 answers
47 views

Let $E=\mathbb{Z}_3[x]/\langle x^2+x+2\rangle$. How many elements of order 2 and 3 are there in the additive group of $E$? How many generators?

I know that for $E=\mathbb{Z}_3[x]/\langle x^2+x+2\rangle $, since $x^2+x+2$ is irreducible over $\mathbb{Z}_3$, $E$ is a field, because $\langle x^2+x+2\rangle $ is a maximal ideal. In addition, $E$ ...
Camilo Diaz's user avatar
1 vote
1 answer
76 views

Galois group of splitting field of $x^3-5$ over $\mathbb F _7$

Honestly, I'm not even sure where to start. I think I understand how to find the Galois group of a field extension with $\textrm{char}\mathbb F=0$ but for some reason I'm confused when it comes to ...
RatherAmusing's user avatar
3 votes
1 answer
76 views

Is every algebraic extension of a finite field Galois?

Let $E/F$ be a (not necessarily finite) algebraic extension, where $F$ is finite. Now, it is known that $E/F$ is a normal extension. On the other hand, $E/F_p$ is algebraic and, since $F_p$ is perfect,...
A Name's user avatar
  • 316
0 votes
2 answers
62 views

In finite fields, generators of $F^*$ under automorphisms in Galois group are also generators

Let $F$ be a finite field with characteristic $p$ and denote $F^*$ the group of invertible elements of $F$. Show that if $a \in F^*$ is a generator, then so is $\sigma (a)$, for all $\sigma \in \...
RatherAmusing's user avatar
0 votes
1 answer
65 views

How many $2\times4$ matrices have nonzero minors in a finite field?

A $2\times4$ matrix $\begin{bmatrix}a_{1,1}&a_{1,2}&a_{1,3}&a_{1,4}\\ a_{2,1}&a_{2,2}&a_{2,3}&a_{2,4}\end{bmatrix}$ has $6$ different $2\times2$ submatrices (the determinants ...
Akiva Weinberger's user avatar
0 votes
1 answer
62 views

Representing the finite field as $\{i*g+j\}$ where $g$ is a generator

This question arose from my thoughts on why the size of a finite field is always a prime power like $p^n$. First, $\Bbb Z/p\Bbb Z$ is a field, and $\Bbb Z/p\Bbb Z -\{0\}$ is a cyclic group under the ...
Hae Koo Jeon's user avatar
1 vote
1 answer
27 views

Confusion regarding standard generator matrix

I think I have a misconception regarding standard generator matrices. Let $G$ be a generator matrix for a code. Then by performing row operations we can put it in reduced row-echelon form. These ...
kubo's user avatar
  • 2,067
1 vote
1 answer
46 views

How to find finite fields of prime power order with two multiplicative subgroups of order $2$ and an odd prime $q$?

I want to find finite fields of prime power order $p^n$ that have only two multiplicative subgroups of order $2$ and a large prime $q$. In particular, I need an odd prime $p$ and an integer $n>1$ ...
Somudro Gupto's user avatar
2 votes
2 answers
85 views

In polynomial quotient rings over $\mathbb{F}_2$, how to use Chinese Remainder Theorem to solve equations?

Suppose I work over the polynomial quotient ring $\mathbb{F}_2[x,y]/\langle x^m+1,y^n+1\rangle$, and I want to solve for polynomials $s[x,y]$ that simultaneously solve the equations \begin{equation} a[...
JoJo P's user avatar
  • 133
0 votes
0 answers
36 views

the multiplication of two field elements in GF((2^n)^2)

While studying GF multiplication in the AES algorithm, I came across the following paper: A new architecture for a parallel finite field multiplier with low complexity based on composite fields In ...
lemoncake's user avatar
1 vote
0 answers
86 views

Number of irreducible polynomials of degree at most n over a finite field

We know that the number $N(n,q)$ of irreducible polynomials of degree $n$ over the finite field $\mathbb{F}_q$ is given by Gauss’s formula $$N(n,q)=\frac{q-1}{n} \sum_{d\mid n}\mu(n/d)q^d.$$ The number ...
Hassen Chakroun's user avatar
1 vote
0 answers
29 views

Order of $\mathbb F _p [x] / (f)$.

I could use some help with the following exercise: Find the number of reducible monic polynomials of degree $2$ over $\mathbb F_p$. Show this implies that for every prime $p$ there exists a field of ...
RatherAmusing's user avatar
2 votes
1 answer
38 views

Let $F \supset K \supset L$ be fields with orders less than 100 and do not include an element $x\neq 1$ that $x^5=1$. Find the order of $F$.

Let $F \supset K \supset L$ be fields with order less than 100 and do not include an element $x\neq 1$ that $x^5=1$. Find the order of $F$. my attempt I used the Tower Law where we have $ [F:L]=[F:K][...
White Give's user avatar
2 votes
0 answers
37 views

uniquness of finite fields if they are inbedded in a algebraic closure

I read in the book of Bosch 3.8 after Theorem 2, that if we fix a algebraic closure of $F_p$ all fields of char p with q elements are equal (not just by isomorphism but really equal). His argument is ...
user1072285's user avatar
2 votes
1 answer
42 views

Artin-Schreier extension is cyclic of degree 1 or $p$

Let $K$ be a field of characteristic $p > 0$ and $K \subset L$ the extension obtained by adjoining the zeros of the Artin–Schreier polynomial $f = x^p − x − a \in K[x]$, where $a\in K^*$, to $K$. ...
math_physics's user avatar
1 vote
0 answers
47 views

Example of an Infinite-Dimensional Non-Commutative Division Ring over a Finite Field

According to Wedderburn's little theorem, any finite-dimensional division ring over a finite field must be a commutative division ring, i.e., it is a field. So the question arises: What about infinite-...
Liang Chen's user avatar
0 votes
0 answers
28 views

What's the point of the local zeta function?

I'm currently reading through Ireland and Rosen's "A Classical Introduction to Modern Number Theory", and I feel I'm missing the point of Chapter 11 on (local) zeta functions. When I see ...
Samuel Johnston's user avatar
0 votes
1 answer
54 views

Basic concepts about irreducible poly in finite fields

I am a bit confused about the behavior of polynomials in finite fields. Why in a splitting field $\mathbb F2[x]/x^3+x+1$,$x^3=x+1$? I have problem in understanding it intuitively, if α is the root, ...
Yi Shen's user avatar
  • 11
0 votes
1 answer
96 views

showing that $x^4+x^3+2$ is primitive over $\Bbb F_3$

I want to show that $x^4+x^3+2$ is primitive over $\mathbb{F}_3$. By definition, this means that $x^4+x^3+2$ is monic and has a root $\alpha$ that generates the multiplicative group of $\mathbb{F}_{3^...
doctor's user avatar
  • 419
3 votes
1 answer
96 views

The rank of Sylow subgroup of special linear groups over finite fields

Let $p,\ell$ be two primes, and let $\mathbb{F}_{q}$ be a finite field of order $q=p^r$. We define the rank of a finite group $G$ to the smallest cardinality of a generating set for $G$. We denote by $...
stupid boy's user avatar
3 votes
1 answer
40 views

Distributing elements of a multi-set to triplets with certain properties possible?

Let $M$ be the multi-set which contains exactly $7$ copies of each positive integer from $1$ to $15$. That is, $M=${$1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,\ldots , 15,15,15,15,15,15,15$}. Is it ...
Stein Chen's user avatar
3 votes
0 answers
108 views

Ideal of multivariate polynomial

Let $p(x_1,...,x_n)$ be an element of $\mathbb{F}_2[x_1,...,x_n]/(x_1^2+1,...,x_n^2+1)$. Is there a way to capture the size or dimension of the ideal $(p(x))$? I would guess that if there is a way, it ...
user 1987's user avatar
  • 774
0 votes
0 answers
41 views

Efficient computation of factorial in finite field

What is the state of the art for fast computation of the factorial function $n!$ and more generally of $\prod_{1\leq x\leq n} (x-q)$ (rising/falling factorial) in a finite field? I found just one ...
Jim's user avatar
  • 538
2 votes
1 answer
50 views

$A^{p^{n!}}-A$ is nilpotent in $M_n(\mathbb{F}_p)$

As mentioned in the title I want to show $A^{p^{n!}}-A$ is nilpotent matrix for any $n\times n$ matrix $A$ with elements in $\mathbb{F}_{p}$. So far I only know if $\lambda$ is eigenvalue of $A$, then ...
Laurence PW's user avatar
1 vote
0 answers
32 views

Inverse function / mapping considering vector multiplication by matrix

Inverse function / mapping considering vector multiplication by matrix also touches symetric encryption Consider, there's a simple matrix as a mapping from R3 ➝ R3 ...
Heinrich Elsigan's user avatar
1 vote
1 answer
61 views

Evaluate the product of (j^n + 1) for j in a finite field

I stumbled across an exercice and this product came up, with the claim: $$\displaystyle\prod_{i \in F_{p}}(i^n+1) = \left\{ \begin{array}{ll} 0 & \mbox{if }\; \dfrac{p-1}{\gcd(p-1, n)} ...
Bij2u's user avatar
  • 104
0 votes
2 answers
81 views

Is this $f(x) = x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$ irreducible in GF(5)?

Perhaps one can somehow apply Eisenstein's sign here by considering $f(x+1)$, but by default it is formulated for the expansion over $\mathbb{Q}$ of a polynomial from $\mathbb{Z}[x]$. Here we have $GF(...
mackenzie's user avatar
0 votes
1 answer
77 views

Show that GF(81) is an $x^{26}+x^{8}+x^{2}+1$ decomposition field

I tried decomposing the polynomial, but after taking out $(x^{2}+1)$ you have to break the remainder into polynomials of degree 4, which is manually hard. Perhaps this is solved by using Frobenius ...
mackenzie's user avatar

1
2 3 4 5
109