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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Rank of matrix over $GF(2)$ whose rows have exactly $k$ elements $1$ [closed]

Consider the $\binom{n}{k}\times n$ matrix $A$ whose rows have $k$ $1$'s and $n-k$ $0$'s. There are no repeated rows. What is the rank of $A$ over $GF(2)$?
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A quadratic solution and its conjugate

Let $\mathbb{F}_q$ be a finite field of $q$ elements such that $q=p^{t}$. Let $\alpha$ be a solution of the equation $Ax^2 +Bx +C=0$ and $\alpha'$ its conjugate, where $A$, $B$ and $C$ are nonzero ...
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Question about some term in Sage while using GF(9)

I tried to define an elliptic curve over $GF(9)$ in Sage, and some term $z2$ appeared, see below (click on the image if the font is too small): I know that it has something to do with the definition ...
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Maximal ideals in $\mathbb{F}_q[x,y]$

Let $p \in \mathbb{P}$ be a prime and suppose that an integer $e > 1$ is given such that the polynomial $s_e = 1 + x + \dots + x^{e-1}$ is irreducible in $\mathbb{F}_p[x]$. My question is the ...
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$A\in \mathbb{F}_q$, then there exists $n\in \mathbb{Z}_{\geq 0}$ such that $A \in \mathbb{F}_{q^n}$ a perfect square.

I was wondering whether the statement in my title holds. I think it does, but I am not sure. I have managed to prove it for the case $A = -3$, but not in the general case. Any ideas or tips?
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Group / field extension solvability in the case of $x^3 - 2$

Whenever a polynomial is solvable by radicals, the Galois group of its splitting field must be a solvable group. A group $G$ is solvable if there are subgroups $H_0, H_1, \dots , H_n$ such that $$\ 1 =...
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Any multiplicative subgroup of a finite field is cyclic

I asked for minimal hints in this question. Now I've come up with a proof. Could you please verify if it is fine or contains logical mistakes? Let $F$ be a finite field and $F^\times = F \setminus \{...
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solution to cube root of finite field

I'm trying to solve the following equation and I'm having trouble $$y^3 = (x)^e \pmod n $$ in my case $x^e = 124205, n = 129071$ I'm expecting this answer to result in $65$ (as this is the thing that ...
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Let $F$ be a finite field. Then the multiplicative group $(F \setminus \{0\}, \cdot)$ is cyclic

Let $F$ be a finite field. I'm trying to prove The multiplicative group $(F \setminus \{0\}, \cdot)$ is cyclic. Then I figure out that it's sufficient to prove Different multiplicative subgroups of ...
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Why is $|\langle A \rangle|$ odd?.

Let $A \in GF(2^r)$. Why is $|\langle A \rangle |$ odd? If $A$ is zero then it only maps to itself, leading the element to have an order of one. Hence, an odd order. Why can I claim for all non-zero ...
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Euler decomposition of symplectic matrix over finite field

Is there a finite field equivalent to the Euler decomposition of symplectic matrices? See previous post for the real version : Finding Euler decomposition of a symplectic matrix
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The order of a finite field

I'm reading a theorem about the order of a finite field: Here is the proof: At the end, the author said It follows that $\mathbb{F}$ is a vector space over $\mathbb{F}_{p}$, implying that its size $...
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How to generate and solve finite fields and insure integers instead of decimals for coefficients and random points

Background: I have tried to follow this tutorial on secret sharing: https://medium.com/@apogiatzis/shamirs-secret-sharing-a-numeric-example-walkthrough-a59b288c34c4. I have managed to use Shamir's ...
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Finite field with 8 elements

I am trying to work out exercise 61 from Rotman's Galois Theory, second edition: 61. Give the addition and multiplication tables of a field having eight elements. (Hint: Factor $x^8 -x$ over $\mathbb{...
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The number of matrices whose rank is $r$ and whose weight is $w$ over a finite field [closed]

Let $M$,$N$ and $m$ be positive integers. Let $p$ be a prime number. Let $w$ and $r$ be nonnegative integers which satisfy $w\leq MN$ and $r\leq \min(M,N)$. We set $q=p^{m}$. Let $\mathbb{F}_{q}$ be a ...
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Factorizing the polynomial $x^6-2x^3+8$ over finite fields

This is a very specific question. This equation came up when attempting to compute certain subgroups of groups of Lie type. The polynomial $x^6-2x^3+8$ splits over $\mathbb{F}_q$ if and only if a ...
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Does $\Phi_n(\alpha)=0$ in $\Bbb{F}_p$ for some $\alpha\in\mathbb{F}_p$ imply that $\mathrm{ord}(\alpha) = n$?

Let $\Phi_n(x)$ denote the $n^\text{th}$ cyclotomic polynomial. Suppose it has a root $\alpha$ in the finite field $\Bbb{F}_p$ and $p \nmid n$. Does it follow that $\mathrm{ord}(\alpha) = n$? In the ...
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Every Galois field $F$ of characteristic $p$ is perfect

I'm trying to do Exercise 2.6.13 from textbook Groups, Matrices, and Vector Spaces - A Group Theoretic Approach to Linear Algebra by James B. Carrell. Could you please confirm if my attempt is fine or ...
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What is the intuition behind mapping of elements from $GF(2^8)$ to $(GF(2)^2)^2)$?

I'm finding it very difficult to understand the concept of mapping elements from the extension field $GF(2^8)$, to $(GF(2)^2)^2)^2 $. I realize that the field that the elements of the field, $GF(2^8)$,...
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How to find the generators of the multiplicative group of a finite field

Input: p (prime number), n (positive number) Output: g ( generator ) I have just found an irreducible polynomial over $F_p[x]$. Now I must find all generators of the multiplicative group from this ...
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Show that $\psi_{b}=\psi_{c}\Leftrightarrow b=c$.

Let $K$ finite field with $|K|=p^{n}$,where $p$ is prime number.Let the transformation (Trace) : $$Tr:K\to K,\ Tr(a)=a+a^{p}+a^{p^{2}}+\cdots a^{p^{n-1}}.$$ For every $a\in K$ we khow it's true that $...
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Why is $\ \frac{1}{2}(p^{rmn-r})(p^r - 3)\ $ odd?

Let $p>2$ be prime and $m,n,r \in \mathbb{Z}^+$. Why is $$\frac{1}{2}(p^{rmn-r})(p^r - 3)$$ an odd number when $p \equiv_4 1$ or when $r$ is even? I'm not really sure how to approach this.
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How to get $(ab)1 = (a1)(b1)$ in Galois field?

I'm reading Galois field from textbook Groups, Matrices, and Vector Spaces - A Group Theoretic Approach to Linear Algebra by James B. Carrell. Here $r,a,b \in \mathbb N$ and $1 \in \mathbb F$. While ...
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Polynomials of degree $n$ with $\Bbb{F}_p$ are always reducible in $\Bbb{F}_{p^n}$

This is a rather basic question, but I can't seem to find any reference here on StackExchange. Is it true that, given a polynomial $p(x) \in \Bbb{F}_p[x]$ of degree $n$, we have that $p(x)$ is always ...
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Extension of $\mathbb{Q}$ by algebraic eigenvalues of commuting operators is finite

Let $K=\mathbb{Q}(a_1,a_2,\cdots)$ be an extension of $\mathbb{Q}$ where the $a_n$ are eigenvalues of an infinite family of operators $T_n$ who commute and whose characteristic polynomials have ...
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The roots of an irreducible polynomial over $\Bbb Z_p$ and a useful equivalence

In the excellent expository papers of Keith Conrad, I stuck at a proof of a proposition in the Finite Fields. Proposition. Let $\pi(X)\in \Bbb Z_p[X]$ be an irreducible polynomial of $\Bbb Z_p[X]$ of ...
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Claim: $f(X)^{p^m}=f(X^{p^m})$, if $f(X)\in \Bbb Z_p[X]$ and $m\in \Bbb N$.

Claim: $f(X)^{p^m}=f(X^{p^m})$, if $f(X)\in \Bbb Z_p[X]$ and $m\in \Bbb N$. Proof 1. Recall that since $R:=\Bbb Z_p[X]$ is a commutative ring of prime characteristic $\mathrm{Char}(R)=p$, then we have ...
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Square values of a polynomial

I need to know if there is an efficient algorithm for knowing the following problem? Consider a monic polynomial $f$ of degree $3$ on the finite field $F_q$ when $q$ is the power of a prime number. ...
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How many elements of order 2 are there in $GF(p^r) \setminus \{0\}$? [duplicate]

I can see why there is only one element in $GF(p) \setminus \{0\}$ of order 2. But I'm not sure why I can claim that this would be the same element and only element for $GF(p^r)$.
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How many elements of $M$ are similar to the following matrix?

Let k be the field with exactly $7$ elements. Let $M$ be the set of all $2\times 2$ matrices with entries in k. How many elements of $M$ are similar to the following matrix? $ \begin{pmatrix} 0 & ...
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How to calculate the irreducible polynomial in galois field

I have a an expression (x^3 + x^2 + 1) / (x^6 + x^5) in GF(2^8) and its primitive polynomials (0,1,3,4,8) How to deal with this ...
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Splitting field of the minimal polynomial

I need a check for this: Let $n=2^p -1$, with $p$ prime and let $\alpha$ a primitive $n$-th root of unity. Show that $\mathbb{F}_{2^p}$ is the splitting field of the minimal poly $M_{\alpha^i}(x)$ I ...
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Independence of coordinate functions on variable.

Consider a permutation $f : (\mathbb{F}_2^{n})^2 \to (\mathbb{F}_2^n)^2$. Let $(x,y) \in (\mathbb{F}_2^n)^2$ be any vector. Write $(z,w) = f(x,y)$. Suppose that I know that $x = 0 \iff z = 0$ and $y = ...
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Equidistribution of powers of primitive roots modulo $p$

Let me start with a nice experimental observation. Fix a large prime, say $p = 5003$. It turns out that $g = 2$ is a primitive root mod $p$. If we plot the powers of $g \in \Bbb F_p^{\times}$ (...
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Relation between generator polynomial and codeword in a cyclic code

I'm trying to solve the following exercise, but I can't use an hypothesis. Let $F$ be a finite field and $a(x)$ a poly of degree $n$ over $F[x]$. Let $C$ the smallest cyclic code of length $n$ over $...
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Let $K = \mathbb{F}_3[T]/(T^3-T+1)$, what would be an irreducible polynomial in $K[X]$ of degree $13$?

Let $K = \mathbb{F}_3[T]/(T^3-T+1)$. I'm trying to find an explicit polynomial $f \in K[X]$ that is irreducible of degree $13$. My first attempt was to note that since $T^3-T+1$ is irreducible over $\...
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Number of irreducible factors of $x^{p^n + 1} - 1$ over $\Bbb F_p$

Let $p,n$ be two odd prime numbers. I want to show that the number of irreducible factors of $x^{p^n + 1} - 1$ over $\Bbb F_p$ is $$N = \frac{p^n-p}{2n} + \frac{p-1}{2} + 2$$ I know that this is equal ...
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Elliptic Curve r-torsion points: Trace and AntiTrace

I am reading Pairings for Beginners from Craig Costello (pdf available for free, just google it) on page 53-55, We consider an elliptic curve E defined on a finite field $F_q$ with q prime q. We are ...
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Finding any elliptic curve equation over any field based on order [closed]

I am looking for a way to find examples of elliptic curves that will have an order that I have been given. The field size cannot be equal to the given order and it needs to work for large numbers. Is ...
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SageMath: defining class of functions on Elliptic Curves

In SageMath, I would like to manipulate rational functions on elliptic curves (defined on finite fields). For example, for $P = (x,y)$ on some curve $E$ $$f = x+y-12$$ $$g = \frac{x+y-3}{(x-3)^2} $$ ...
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Number of $\alpha\in\Bbb{F}_{27}$ such that $|A_{\alpha}|=26$ equals $12$ [duplicate]

Question: Let $\Bbb{F}_{27}$ denote the finite field of size $27$. For each $\alpha\in\Bbb{F}_{27},$ we define $$A_{\alpha}=\{1,1+\alpha,1+\alpha+\alpha^2,1+\alpha+\alpha^2+\alpha^3,\dots\}.$$ Then ...
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1answer
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If a polynomial is irreducible and nonconstant over a finite field, it has a multiple root iff it is in the variable $x^p$

I am a very basic field theory question. I must be mixing up a theorem here, but I am unsure which. My goal here is to determine if there exists an inseparable, irreducible polynomial in a finite ...
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1answer
37 views

Degree of a multivariate polynomial over a finite field with many roots

Question Let $q$ be a prime power, $k\in\{1,\ldots,q-1\}$ and $f$ be a multivariate polynomial in $\mathbb{F}_q[x_1,\ldots,x_n]$ having $q^n - k$ roots. Show that $\deg(f) \geq (q-1)n - k + 1$. (The ...
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Find idempotent given generator poly and check poynomial by Bezout algorithm

I have the cyclic code $C$ of length $8$ and dimension $4$ over $\mathbb{F}_3$ and with check polynomial $$g(x) = (x-\alpha)(x-\alpha^2)(x-\alpha^3)(x-\alpha^6) = x^4+x^3+x+2$$ where $\alpha \in \...
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Canonical forms of matrices under congruence relation

Let $A,B$ be square matrices over a field $K$ (We can assume $K$ to be finite, if needed). Consider the equivalence relation $A\sim B$ if and only if there exists an invertible matrix S such that $A=...
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35 views

Galois groups of polynomial above a field of 17 elements

Let f(x) be polynomial with a degree 4, which is a separable and irreducible. What are the possible Galois groups above a field with 17 elements of f(x)? I am a little confused, since 17 is a prime ...
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31 views

Group cohomology of Galois group of finite extension of finite fields

Let $E/F$ be a finite extension of finite fields; hence, it is a cyclic Galois extension, so let the Galois group be $G$. Hilbert's Theorem 90 states that $H^1(G, E^{\times})=0$. My question is: How ...
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Uniqueness of linear codes

In this textbook, I found the following remark: An $(n,k)$ linear code $\mathcal{C}$ is a unique subspace consisting of a set of $2^k$ codewords. The statement surprised me because in vector spaces ...
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1answer
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SageMath: defining an extension of a Finite Field

I am trying to do basic 101 manipulation with SageMath F = GF(3); F Finite Field of size 3 R.<x> = F[] ; R Univariate ...
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1answer
37 views

Order of elliptic curve $y^2 = x^3 + ax^2 + b^2x$ is multiple of $4$.

Let $\mathbb{F}_q$ be a finite field with odd characteristic and let $a,b \in \mathbb{F}_q$ with $a \neq \pm 2b$ and $b \neq 0$. Define the elliptic curve $E: y^2 = x^3 + ax^2 +bx$ The goal is to show ...

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