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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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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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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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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Syndrome decoding algorithm and standard form

Assume I have a linear code over $\mathbb{F}_2$ with dimensions $[3,6]$ and generator matrix $G$ not in standard form $$\begin{bmatrix} 0&1&0&1&1&0 \\ 1&0&0&1&0&...
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Determine if a $4$-th root of unity is contained in $\mathbb{F}_9$

I have two questions, one of those is the same here, but I'd like to use another argument and I need a check! The text is: i) Is it true that a primitive 3-th root of the unit over $\mathbb{F}_3$ is ...
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Decomposition of symmetric matrices over $\mathbb{F}_2$

Can every $n\times n$ symmetric matrix over $\mathbb{F}_2$ be decomposed into $\sum_{v=1}^k v_i v_i^T$ for vectors $v_1,\ldots,v_k\in\mathbb{F}_2^n$ and integer $k$? As far as I know, for symmetric ...
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Are the $n$-th roots of unity over an arbitrary field generated by a single element?

Let $\mathbb F$ be a field. If we take the set of numbers such that $x^n=1$ (for a fixed $n$), is it true that said group is finitely generated? What I mean is, let $n\in\mathbb Z^+$, $G_n(\mathbb F):=...
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Find inverse element of $1+2\alpha$ in $\mathbb{F}_9$

Let $$\mathbb{F}_9 = \frac{\mathbb{F}_3[x]}{(x^2+1)}$$ and consider $\alpha = \bar{x}$. Compute $(1+2 \alpha)^{-1}$ I think I should use the extended Euclidean algorithm: so I divide $x^2 +1 $ by $(...
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Construct a Reed Solomon code: find the parity check matrix

I am trying to solve the following exercise, but I need a check/opinion on how to solve it. Construct a Reed-Solomon code with dimensions $[12,7]$ over $\mathbb{F}_{13}$ and find a parity check ...
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Isomorphism of the fields $\mathbb{Z}[i]/p\mathbb{Z}[i]$ and $\mathbb{F}_{p^2}$

I am given the task of showing that $\mathbb{Z}[i]/p\mathbb{Z}[i]\cong\mathbb{F}_{p^2}$ for $p\equiv3$ mod $4$ prime. I understand that there exists only one field of order $p^2$ up to isomorphism, so ...
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Solving $aX^3 + bY^3 + cZ^3 - dXYZ = 0$ over $\mathbb{F}_q$

I am looking for a way to solve the equation mentioned in the title with $Z\neq 0$ over the finite field $\mathbb{F}_q$ without going through all $q^3$ possibilities. I was thinking: maybe we can ...
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Why are there $p+1$ solutions to a projective line over a finite field of order $p$

Let $\mathbb{F}_p$ be a finite field with $p$ elements, and let $$x+y+z=0$$ be a projective line with $x,y,z \in \mathbb{F}_p$. In a book I am currently reading about elliptic curves, it uses the fact ...
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How does one get this generator matrix when it comes to finite fields and codes?

For binary messages of length $9$ we define code words as follows: We write the letters one by one into the rows of a $3 \times 3$ matrix, which we enhance by a fourth row and column to a $4 \times 4$ ...
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True or False questions regarding $𝔽_9$ with the irreducible polynomial $x^2 +2x+2$

Let $𝔽_9$ be constructed with the irreducible polynomial $x^2 +2x+2$. For $a,b \in 𝔽_3$ we write $ax+b \in 𝔽_9$ for $ab$. In our exam we had to find out whether the following are true or false. I ...
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Why is $x^4+x^2+1$ over $𝔽_2$ a reducible polynomial? What do I misunderstand?

I don't quite understand when a polynomial is irreducible and when it's not. Take $x^2 +1$ over $𝔽_3$. As far as I know, I have to do the following: 0 1 2 using $x \in 𝔽_3$ 1 2 2 using $p(x)$ I ...
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Find relationship between $1+x$ and generator polynomial

I can't solve the following exercise and I need a help. Consider $\mathcal{C}$ binary cyclic code with length $n$ with generator polynomial $1 + x$. Let $\mathcal{C'}$ be the binary cyclic code of ...
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Primitive elements in fields and finite fields

I have the following two definitions: If $K$ is an extension field of $F$ and $K = F(a)$ for some $a \in K$, then $a$ is a primitive element of $K$. If $K$ is a finite field and $a$ is a generator ...
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For $A \in F_2^{N \times M}$ find $\mathbf x \in F_2^M$ to maximize $L_0$ norm of $A \mathbf x$

I have an optimization problem: For a fixed matrix $A \in F_2^{M \times N}$ find a vector $\mathbf x \in F_2^{N \times 1}$ such that the vector $\mathbf y = A \mathbf x \in F_2^{M \times 1}$ has ...
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The equation $x^2-x-1$ has no solution over finite fields of even order.

Is the equation $x^2-x-1$ has no solution over $GF(2^i)$ for all i. I can prove it trivially for some arbitrary chosen small even ordered fields, but can this be generalized?
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When a finite extension $E$ of a finite prime field $F$ can be linear spaned by a subset of $E$?

Let $F$ be a finite field of order $p$, and $E$ be an extension of $F$ with $[E:F]=2n$, then $|E|=p^{2n}:=q$. Now denote $\Omega=\{ x\in E:x^{\sqrt{q}+1}=-1\}$. My question is: (1) Is it true that $\...
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Proving $z^2 = 2$ mod $p$ where $ p = 1$ mod $8$ [duplicate]

I am following a course in algebra and I am asked to prove that, if we take any prime $p$ such that $p = 1\bmod 8$, $z \in \Bbb Z$ exists, such that $z^2 = 2\bmod p$. I have proven that the order of $...
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Show a code is cyclic given the generator matrix

I am given $\mathcal{C}$ the following code over $\mathbb{F}_2$ with generator matrix $$\begin{bmatrix} 0 & 1 & 0 & 1 & 1 &0 \\ 1 & 0 & 0 & 1 & 0 &1 \\ 0 & ...
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Dimension of the field extension $F[x]/(f)=\deg(f)$ for an irreducible monic polynomial $f\in F[x]$

Let $F$ be a field and let $f$ be an irreducible monic polynomial in $F$. Prove that the dimension of $F[x]/(f) = \deg(f)$. Here is how I am proceeding. Let $$f(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{...
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Let $ \mathbb{K}$ be a field with 27 elements and $ a \in \mathbb{K}$. Then there exists $ x, y \in \mathbb{K}$ such that $ x^3 + y^5 = a$.

Let $\mathbb{K}$ be a field with 27 elements and $ a \in \mathbb{K}$. Then there exists $ x, y \in \mathbb{K}$ such that $ x^3 + y^5 = a$. How to try? Using extensions fields?
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Schoof Algorithm : working on an example with SageMath

I am a noob amateur interested in Elliptic Cryptography and I am trying to work on Schoof Algorithm on a small example with the help of Sagemath the algorithm description i found in a pdf called "...
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Equivalence of two cyclic codes

I'm trying to solve the following, but have no clue: Let $$\sigma: \{0, \ldots, n-1\} \rightarrow \{0,\ldots,n-1 \}$$ $$x \mapsto x+a \text{ mod} n$$ a permutation with $(a,n)=1$. Assume to have $\...
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Find decomposition of $x^8 -1$ over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_4$

Let us consider $x^8 -1$. I want to decompose it over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_4$. In $\mathbb{F}_3$ I have no problem, since I can use the cyclotomic cosets. In $\mathbb{F}_2$...
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Polynomials of odd degree over $F_{p}$ have a root in $F_{p}$

I'm reading Broker's paper "CONSTRUCTING SUPERSINGULAR ELLIPTIC CURVES", which gives an algorithm of constructing supersingular elliptic curves over $F_{p}$ (where $p$ is a prime number). Theorem ...
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Prove that such a code does not exist

I'm trying to solve the following exercise, but don't know what results to use. Prove that it is not possible to find a linear code $\mathcal{C}[8, 5, 4]$ over $\mathbb{F}_2$ (without using the ...
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How does the polynomial $X^{p-1}+1$ split over $\mathbb{F}_p$

Is there a well-known formula for the irreducible factors of the polynomial $X^{p-1}+1$ over $\mathbb{F}_p$ where $p$ is an odd prime? Thanks in advance.
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Find splitting field and irreducible decomposition [duplicate]

I'm trying to solve the following exercise. I did the first two points( hope they're right), but have no idea on how to solve the last one. Let $f(x)= x^8 -x$ in $\mathbb{F}_3[x]$. Find: ...
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Square root in $\mathbb F_{2^n}$

Let $\mathbb F_{2^n}$ be a finite field with $2^n$ elements. I am just wondering if it is true that for all $n\in \mathbb N$ all elements of $\mathbb F_{2^n}$ have square roots, i.e for all $a\in \...
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$E(\mathbb{F}_q ) \cong \mathbb{Z}/n \mathbb{Z}×\mathbb{Z}/n \mathbb{Z}$ implies $n|q-1$

Let $E$ be an elliptic curve over $\mathbb{F}_q$ with $E(\mathbb{F}_q ) \cong \mathbb{Z}/n \mathbb{Z}×\mathbb{Z}/n \mathbb{Z}$ for some $n \in \mathbb{N}$ I want to show that $n$ divides $q − 1$ and ...
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Prove that solution of equation $x^p=a$ exists, where $a$ is a fixed element in a finite field $F$ and $charF=p.$

I have to prove that solution of equation $x^p=a$ exists and it's unique, where $a$ is a fixed element in a finite field $(F,+,\cdot,0,1)$ and $charF=p.$ I know how to prove that $p$ is a prime number ...
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polynomials in $\mathbb F_p$

Let $p$ be a prime number and denote by $\mathbb F_p$ the field of integers modulo $p$. Like before, we can consider polynomials with coefficients in $\mathbb F_p$. Then, the polynomial $X^p − X$ is ...
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When swapping rows in a matrix, what happens to the numerical range?

What effect does the operation of swapping rows in a matrix have on the matrix's numerical range? Whatever the result is, is it the same for matrices over finite fields? Thanks, Derek
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Polynomial with roots modulo all primes $p \equiv 3 \pmod 4$

Does there exist an irreducible polynomial $f \in \mathbb{Z}[X]$ of degree $n \geq 2$ with a zero modulo all primes $p \equiv 3 \pmod 4$? For example, there is such a polynomial $X^2+1$ if we choose ...
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Show that if $p,q$ prime, $p<q$, and $p\not\mid q-1$, then there is $L:\mathbb{F}_q$ which is a splitting field for each $x^p-a,a\in\mathbb{F}_q^*$.

I'm working my way through the exercises in a book on Galois theory. Right now I've got one exercise left in the chapter on finite fields before I continue to the next chapter. But for this one, I ...
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Is an integral domain $\frac{\mathbb{C}[x,y]}{<x^4+x^3y+y^4>}$?

Is an integral domain $\frac{\mathbb{C}[x,y]}{<x^4+x^3y+y^4>}$ ? Where $\mathbb{C}[x,y]$ is a commutative ring of polynomials over $\mathbb{C}.$ I know the fact that for any field, $\mathbb F[...
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Proving no matrix of given order can exist over finite field

I was asked to show in an exercise that for a prime $ p $, in the group $ GL(n,F_p) $ of invertible matrices of dimension $ 1 \leq n \leq p $ mod $ p $, that no matrix can have order $ p^2 $, meaning ...
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Understandig proof of a theorem on finite fields.

The theorem statement is: a finite field of characteristic $p$ has $p^n$ elements. I found this very simple proof in the book "Ling, San; Xing, Chaoping; Coding Theory - A First Course". But I don't ...
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Restriction of the Frobenius automorphism for normal extensions

I'm studying number theory on Marcus book and at a certain point I'm required to prove the following facts about the Frobenius automorphism. We start with a lemma and then are required to specialize ...
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Minimum distance for a cyclic code

I am given the code $C[8,4,d]$ over $\mathbb{F}_3$ with generator polynomial $g(x)=x^4 + 1$. From theory I know that the check polynomial is given by the division of $x^8 - 1$ by $x^4+1$. This gives ...
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Procedure to find explicit isomorphisms between two quotient rings: $\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{R}[x]/(x^2 + \alpha^2)$

Given a ring homomorphism, I'm pretty comfortable with proving whether or not it is an isomorphism. However, I'm having trouble figuring out how to systematically coming up with a ring homomorphism in ...
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Idempotent and generator polynomial

I need a check on the following exercise. Let $C$ be the code over $F_2$ of length $7$, whose idempotent polynomial is $e(x)=1+x^3+x^5 + x^6$. Find its generator polynomial $g(x)$ and use the BCH ...
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Generator matrix of twisted Gabidulin codes

If we consider twisted Gabidulin codes proposed by Sheekey as follows: Let $n, k, s$ be positive integers such that $k<n$ and $\gcd(s, n)=1$. Let $\eta$ be a nonzero element in $\mathbb{F}_{q^...
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1answer
24 views

Average value of the orders of all elliptic curves over the finite field of p-elements

Is true that the average value of the orders of all elliptic curves over $\mathbb F_p$ is $p+1$? More precisely, fix a prime $p$ and let $\mathbb F_p$ be the field of $p$ elements. Consider the set $...
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43 views

Symmetric Matrix over a finite field of Characteristic 2

Let $M$ be a $n$ by $n$ symmetric matrix over a finite field of Characteristic 2. Suppose that the entries in the diagonal of $M$ are all zero, and $n$ is an odd number. I found that the rank of $M$ ...
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List all monic irreducible polynomials of prime degree $p$ over $\mathbb{F}_p$

There are $p^{p - 1} - 1$ monic irreducible polynomials of prime degree $p$ over $\mathbb{F}_p$ by this post. The chance of picking one of them randomly is $\cfrac{p^{p - 1} - 1}{p^p} = \cfrac{1}{p} -...
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Quadratic Integer Ring mod p - Field?

In order to understand a proof of a book I try to get I need to deal with Quadratic Integer Rings. As far as I got till now if I look at $\mathbb{Q}(\sqrt(d))$, $O_{\sqrt(d)}$ and $p \equiv \eta_p \...

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