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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 $\operatorname{GF}(q)$.

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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
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2 votes
1 answer
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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
37 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
0 votes
1 answer
100 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
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1 vote
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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
43 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
-3 votes
0 answers
26 views

Irreducible polynomial over a finite field is irreducible in $\mathbb{Z}$. [closed]

Let $f\in\mathbb{Z}[x]$ be a monic polynomial of degree 5. Furthermore suppose that $p$ is a prime number and that $F$ is a finite field of order $p^2$ such that $f$ has no roots (in $F$). Show that $...
Thora N's user avatar
  • 67
1 vote
1 answer
69 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
2 votes
0 answers
56 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
  • 306
0 votes
2 answers
57 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
61 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
55 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
0 votes
0 answers
19 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
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1 vote
1 answer
42 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
82 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
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0 answers
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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
84 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
1 vote
0 answers
35 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
37 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
43 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
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0 answers
25 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
48 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
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
85 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
101 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
34 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
49 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
57 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
78 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
74 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 vote
1 answer
53 views

Why $\sum\limits_{x, y \in \mathbb F_q} (x+y)(x-y) = \sum\limits_{a, b \in \mathbb F_q} ab$

Let $q$ be a prime power and $\mathbb F_q$ the finite field with order $q$, why $\sum\limits_{x, y \in \mathbb F_q} (x+y)(x-y) = \sum\limits_{a, b \in \mathbb F_q} ab$? We can denote $a = x+y$ and $b=...
soda's user avatar
  • 33
3 votes
0 answers
66 views

Is this connection between prime numbers, prime polynomials, and finite fields true?

I recently learned of the following connection between prime numbers and prime polynomials in the field of cardinality $2$. Namely, you take a natural number $n$, and use the digits of the base $2$ ...
user107952's user avatar
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1 vote
0 answers
27 views

Automorphism of a character and Frobenius morphism

Let $T$ be a torus, and let $X(T)$ be its characters group (the group of homomorphisms $T \to \mathbb{G}_m$). I am trying to understand the proof of the following result (Proposition 4.2.3) in Digne-...
Conjecture's user avatar
  • 3,260
0 votes
0 answers
38 views

Given a singular matrix $B$ and a result $C=A\times B$, find matrix $A$ over finite fields.

The problem is a conituation of this problem, but over finite fields. In the context of a finite integer field, particularly when all entries in matrices $A, B$, and $C$ are drawn from a finite ...
X.H. Yue's user avatar
2 votes
1 answer
70 views

Why should a splitting field of a polynomial over $\mathbb{F}_p[X]$ be a cyclotomic extension?

I have just been doing a Galois theory exercise and one part of the exercise requires me to explain why, if $f \in \mathbb{F}_p[X]$ is a polynomial of degree $n$, its splitting field $L$ is a ...
Featherball's user avatar
1 vote
3 answers
94 views

Find minimal polynomial over $\mathbb{F}_p$ of a generator of $\mathbb{F}_{p^n}^{*}$

So this is part of an exercise sheet where I thought I had figured it out but turns out I didn't. Theory from lecture: We know for a prime $p$ the finite field $\mathbb{F}_{p^n}$ is isomorphic to $\...
arridadiyaat's user avatar
0 votes
0 answers
29 views

Fermat's little theorem in $F_p(x,y)$

I'm currently starting to study finite fields and I'm wondering if Fermat's little theorem always apply. We know that $a^p \equiv a$ in $\mathbb{F}_p$, but f.e. is it true that $x^p \equiv x$ in $\...
Wicowan's user avatar
  • 75
2 votes
1 answer
27 views

Finite inversive planes and $PSL_2(q)$

Example 6.2.4 of Dixon and Mortimer's, Permutation groups introduces inversive planes as Stainer systems. The classical (real) inversive plane can be naturally understood as a one-point extension of ...
Antonio Montero's user avatar
0 votes
0 answers
24 views

Degrees of irreducible polynomials over finite fields [duplicate]

Artin chapter 15 states the following as a corollary: Corollary 15.7.4 For every positive integer $r$, there exists an irreducible polynomial of degree $r$ over the prime field $\mathbb{F}_p$. ...
Ben Carpenter's user avatar
0 votes
0 answers
53 views

When is $f = X^4 -1 \in \mathbb{F}_p[X], p $prime, irreducible and/or seperable? [duplicate]

I'm having some trouble figuring out a solution to this. I understand that $f$ is separable, iff all its roots are distinct, however I'm completely clueless about how to investigate that criterion......
Raiden's user avatar
  • 17
1 vote
1 answer
44 views

For the Frobenius automorphism, must the fixed field be prime order?

Does it make sense to discuss the Frobenius automorphism when the fixed field has prime power order? For example GF(16)/GF(4) or GF(81)/GF(9)? I can't see anything immediately wrong, but I've never ...
NewViewsMath's user avatar
-2 votes
1 answer
105 views

Solving linear systems in Finite fields [closed]

Consider the following matrix in image. How to find a set of 16 integers with no repetition in [0, 31] such that applying the matrix to this set written as a column vevtor gives the remaining 16 ...
Mathslover shah's user avatar
0 votes
1 answer
49 views

inverse image of Trace mapping from $\mathbb F_{q^m}$ to $\mathbb F_{q}$

Let $T: \mathbb F_{q^m} \to \mathbb F_{q}$, $T(\alpha)=\alpha + \alpha^{q} + \cdots + \alpha^{q^{m-1}}$. It is easy to check that $T$ is epimorphism. The question: for each element $a \in \mathbb F_q$,...
soda's user avatar
  • 33
1 vote
0 answers
49 views

$\operatorname{PGL}_2(\mathbb F_q)$-Stabilizer size of a representation of a finitely generated group

Let $\mathbb F_q$ be a field of $q$ elements where $q$ is an odd prime power. Let $G$ be a finitely generated group and $\rho:G \to \operatorname{GL}_2(\mathbb F_q)$ be an irreducible representation ...
Conjecture's user avatar
  • 3,260
1 vote
0 answers
46 views

If I do random walks on recurrence relations over $\mathbb F_2$, what do I get? Some kind of discrete version of stochastic calculus?

Context: The other day I investigated a special case of recurrence relations that was an iterated running xor on bit streams which I had tried connecting to differential operators and differential ...
mathreadler's user avatar
5 votes
3 answers
88 views

Bijectivity of $f(x) = α^qx + αx^q$ over $\mathbb{F}_{q^3}$

Let $q$ be an odd prime power and $α\in \mathbb{F}_{q^3}$ nonzero. Define the map $$f(x) = α^qx + αx^q.$$ I am supposed to show that $f: \mathbb{F}_{q^3} \to \mathbb{F}_{q^3}$ is bijective. My ...
Tobius's user avatar
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