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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13 views

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
19 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 $...
2
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1answer
38 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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35 views

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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20 views

Determine degree min. polynomial

I need a check on the following question Let $\alpha$ a primitive element of $\mathbb{F}_{2^n}$. Determine the degree of the minimal polynomial over $\mathbb{F}_2$. What can you say about the ...
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26 views

Smallest field and root of unity

I'm trying to solve the following: Let $K$ be the smallest field, with characteristic $2$ such that it contains a $15$-primitive root of the unit. Find its cardinality and a primitive element of ...
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1answer
23 views

Elements of $\mathbb{F}_7^*/\mathbb{F}_7^{*3}$

I think I have forgotten some basic group theory, but I am having hard time representing the elements from $\mathbb{F}_7^*/\mathbb{F}_7^{*3}$, where $\mathbb{F}_7^{*3}$ denotes all elements that are ...
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4answers
65 views

why does $P(X) = X^3+X+1$ have at most 1 root in $F_p$?

why does $P(X) = X^3+X+1$ has at most 1 root in $F_p$ ? I could fact check this on Sage for small values of $p$. For example $p=5$ or $7$ or $19$; there is no root. If $p = 11, 2$ is the only ...
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1answer
28 views

Supersingular Elliptic Curves

I am an amateur and a beginner in the topic A theorem states Let K be a field of characteristic p an elliptic curve is supersingular iff $card(E(K)) = 1$ mod $ p$ supersingular means: the ...
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40 views

Primitive third root of the unit

I'm struggling with this question: Let us consider $\mathbb{F}_9$. Is it true that a primitive 3rd root of the unit over $\mathbb{F}_3$ is contained in $\mathbb{F}_9$? I just know that $x=1$ is a ...
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43 views

How quickly find we find a primitive root of unity in $\mathbb{F}_{p^z}$?

If we are working in a finite field of integers adjoined with $z$ values, we have $\mathbb{F}_{p^z}$, assuming that we constructed the field correctly. How quickly can we find a value, $\omega$, that ...
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1answer
31 views

Finding irreducible polynomial in finite field

I would like to find an irreducible polynomial of degree $3$ in $\mathbb{F}_4$, where $$\mathbb{F}_4 = \{a+b\alpha| \ a, b\in \mathbb{F}_2, \alpha^2 = \alpha + 1\}.$$ I first tried to find an ...
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3answers
47 views

Number of elements $a\in\mathbb{F}_{5^4}$ such that $\mathbb{F}_{5^4}=\mathbb{F}_5(a)$

Determine the number of elements $a\in\mathbb{F}_{5^4}$ such that $\mathbb{F}_{5^4}=\mathbb{F}_5(a)$, and find the number of irreducible polynomials of degree $4$ in $\mathbb{F}_5[x]$. My thoughts: ...
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0answers
56 views

A Curve Over Finite Field

Let $f(x)\in \mathbb{F}_q[x]$ and let $X$ be the curve over $\mathbb{F}_q[x]$ defined by $\psi(x,y)=(f(x)-f(y))/(x-y)$. I want to show that if $(a,b)$ is a simple rational point of $X$ then the ...
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3answers
42 views

$1^{-1}+2^{-1}+\dots+\Big(\frac{p-1}{2}\Big)^{-1} \equiv -\frac{2^p - 2}{p} \mod p$ for an odd prime $p.$

I've reduced a problem down to proving this identity. Unfortunately, I don't know where to even start. There has to be some way of expanding the RHS or combining terms on the LHS, but I don't see it. ...
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1answer
29 views

$Q\subset L$ with $G := \text{Gal}(L/Q)$, Is $L$ contained in the field of constructible numbers? [closed]

$Q \subset L$ is a finite Galois extension with $G := \text{Gal}(L/Q)$ and $G$ is isomorphic to $S_3$, the symmetric group on $3$ elements. Is $L$ contained in the field of constructible numbers?
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1answer
39 views

Minimal extension field of $\mathbb{F}_2$ such that

Find the minimal extension field of $\mathbb{F}_2$ such that this extension contains an element of order $21$? Attempt: I know that such an extension of $\mathbb{F}_2$ is like $\mathbb{F}_{2^s}$ and $...
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57 views
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The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$

Suppose $\mathbb{F}=\mathbb{F}_{q^2}$, where $q$ is a prime power. The conjugate of elements in $\mathbb{F}$ is defined by $\overline{x}=x^q$. I need to find the number of $n\times n$ unitary ...
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Find splitting field of $(x^3-x^2-x)(x^4-x^2+1)$ over $\mathbb{F}_3$

As written in the title, I have to compute the splitting field of $$(x^3-x^2-x)(x^4-x^2+1)$$ over $\mathbb{F}_3$ I'd like to understand if my attempt is correst, or if I'm missing something. Here's ...
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1answer
39 views

Find irreducible factors without factorizing [closed]

I have an exercise from my course notes that states: Find how many irreducible factors has $f(x) = x^{26}-1$ over $\mathbb{F}_3$ and their degrees. (don't factorize it) I see immediately that the $...
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1answer
26 views

Number of Elliptic Curves over Fp

I am a beginner/amateur in the topic according to https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf on page 45, There are approximately 2p different elliptic curves defined over ...
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30 views

Given eigenvectors, what is the set of eigenvalues that makes the matrix have specific coefficients?

I have a $\mathbb{F}_2^{n\times n}$ matrix $P$ = $ \begin{bmatrix} 0 & 0 & \ldots & 0 & -c_0 \\ 1 & 0 & \ldots & 0 & -c_1 \\ 0 & 1 & \ldots & 0 & -c_2 \\...
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1answer
26 views

Is there an isomorphism of fields between $\mathbb{F}_{3^{2}}$ and $\mathbb{F}=\{a+bi; a,b \in \mathbb{F}_{3}\}$?

if $\mathbb{F}=\{a+bi; a,b \in \mathbb{F}_{3}\}$ where $i=\sqrt{2}=\sqrt{-1}$ and we define $(a+bi)+(c+di):=(a+c)+(b+d)i$ and $(a+bi)\ast (c+di):=(ac-bd)+(ad+bc)i$ Is there an isomorphism of fields ...
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1answer
62 views

How to find the inverse of elements in a field?

Normally I use the naive method: $$a^{-1} = a \cdot b \bmod p \equiv 1,$$ where b is the inverse of a. Else I love to use Fermat's little theorem: $$a^{p − 1} \equiv 1 \bmod p.$$ By multiplying both ...
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1answer
11 views

Do diagonal elements in the Galois Field addition tables have to be zeros?

I've seen addition and multiplication tables for Galois Fields, where the addition table is simply modular arithmetic, and some tables where the diagonal elements are zeros (i.e. the additive inverse ...
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3answers
58 views

Proving F is a field [closed]

Let $F=\{a+bi; a,b \in F_3\}$ where $i=\sqrt{2}=\sqrt{-1}$ and we define $(a+bi) + (c+di) := (a+c)+(b+d)i$ and $(a+bi) * (c+di):= (ac-bd)+(ad+bc)i$ with $0=0+i$ and $1=1+0i$ Prove $F$ is a field
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1answer
66 views

Is it possible to define trignometric functions over a finite field? [closed]

Is it possible to define trigonometric functions such as sine, cosine, etc modulo a prime p?
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1answer
36 views

How to construct a field with 25 elements of a given polynomial? [duplicate]

Let us say that the polynomial is $x^2 + 5$ and the field is $\mathbb F_{25}$. Hereby $ax+b$ denotes any element of $\mathbb F_{25}$ with both $a$ and $b$ in the field $\mathbb F_5$.
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45 views

How to show that a polynomial is irreducible

How do I show that a polynomial is irreducibel? How do I show that $x^2+1$ is irreducible over the field $F_p$ where $p \equiv 3 \mod 4$? My guess for number 1) is that inserting all numbers $x$ from ...
3
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0answers
52 views

How many vectors $w\in V$ such that $v_1,…,v_d,w$ are linearly independent?

Let $K$ be a field with $|K|=q$ elements and let $V$ be a $K$ vector space. If $v_1,...,v_d$ linearly independent in $V$. How many vectors $w\in V$ are there such that $v_1,...,v_d,w$ are linearly ...
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1answer
20 views

Is there an analog of Sturm sequences for finite fields?

In finite fields, is there anything analogous to Sturm sequences for counting the number of roots of a polynomial in a given interval? Alternatively, showing that there are zero roots in a given ...
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2answers
42 views

Finite field, basis

In $\mathbb{F}_3^3$, I am given:$$U = \text{span}\left(\begin{pmatrix}0\\1\\2\end{pmatrix}, \begin{pmatrix}1\\1\\1\end{pmatrix}\right),\quad W = \text{span}\left(\begin{pmatrix}-1\\0\\3\end{pmatrix}, \...
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1answer
181 views

What is the explanation for why a field cannot have certain values like e.g. 12? [duplicate]

Ok, as far as I understand a field has to look like $\mathbb{F}_{p^n}$. But why? What is the explanation?
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1answer
26 views

Compute the inertial degrees of two prime ideals e.g the inertial degree of $P/(2)$ where P is prime in $Q[e^{\frac{2\pi i}{23}}]$ lying over (2))

I was reading through Marcus Number Field chapter 3 and I got stuck on exercise 17 Let $K=\mathbb{Q}[\sqrt{-23}]$, $L=\mathbb{Q}[\omega]$ where $\omega=e^{\frac{2\pi i}{23}}$. We know that K $\...
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1answer
76 views

Demonstrate that $p(x)=q(x)$

I want to demonstrate the following statement. Let $p,q\in \mathbb{GF}_2[x_1,\ldots,x_n]$ be of degree $n$ such that for all $v_1,\ldots,v_n\in\mathbb{GF}_2$, $p(v_1,\ldots,v_n)=q(v_1,\ldots,v_n)$. ...
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1answer
77 views

Why this polynomial reducible? (composite field)

In galois field of prime 2, in composite field $GF((({2}^2)^2)^2)$, There are irreducible polynomials and reducible polynomials. $GF(2^2):Q_1(x) = x^2+x+1,$ $GF((2^2)^2):Q_2(x) = x^2+x+\phi,$ $\...
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2answers
36 views

(Proof verification) there is no homomorphism between a finite field’s additive group to its multiplicative group.

Given a finite field F with additive group $\text{F}^+$ and multiplicative group $\text{F}^{\times}$ Show that there doesn’t exist $f$:$\text{F}^+ \to \text{F}^{\times}$ s.t. $f(x+y)=f(x)f(y)$. Proof ...
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1answer
25 views

Minimal size of a sumset over $\mathbb{F_p}$

Let $A, B \subseteq \mathbb{F_p}$ (p prime). How to show that $|A+B| \ge \min\{p, |A|+|B|-1\}$? Since $\mathbb{F_p}$ has only $p$ elements, $\forall S \subseteq \mathbb{F_p}, |S| \ge \min\{p, |S|\}$. ...
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2answers
87 views

$\mathbb{Z}$ mod $p$ vs. $\mathbb{Z}_p$

What is the difference between working in $\mathbb{Z}$ mod $p$ and working $\mathbb{Z}_p$? I'm mainly interested in the terminology and nomenclature, I understand that the result would be the same. ...
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1answer
44 views

Calculate order of multiplicative group of finite field

How can one calculate the order of a multiplicative group of a finite field such as: $(\mathbb{F}(2^3) \backslash \{0\}, \times)$ Is it as simple as doing $2^3-1$ ?
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0answers
18 views

Reed Solomon step by step decodification with an example.

Assuming that we have a RS code with parameters m=5, t=3 defined over the GF(32) with generator poly: ...
3
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3answers
77 views

Roots of $x^{p^{n-1}}+\ldots+x^p+x$ in $\mathbb{F}_{p^n}$

Let $\mathbb{F}_q$ denote a field with $q=p^n$ elements, where $p$ is prime. Consider the polynomial $f=x^{p^{n-1}}+\ldots+x^p+x$ and the sets $$ \begin{align*} S&=\{a^p-a:a\in\mathbb{F}_q\},\\ ...
3
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1answer
25 views

Absolute value of sum of additive characters of $\mathbb{F}_p$

Consider the absolute value of the following exponential sum: $\left|\sum_{x \in \mathbb{F}_p} \sum_{y \in \mathbb{F}_p}e^{\frac{2\pi i}{p}(ux+vy-wxy)}\right|$ for given $u,v,w\in\mathbb{F}_p$ with ...
1
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1answer
98 views

Arithmetic in GF$(2^{32})$ using GF$(2^{16})$ and extensions

Ultimately, I'm looking to implement arithmetic in GF$(2^{32})$. I have a library that implements arithmetic in GF$(2^{16})$ using look-up tables for log and anti-log to implement multiplication, and ...
-1
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1answer
38 views

Use of irreducible polynomial in finite field construction

When constructing a finite field $\mathrm{GF}(p^n)$ using polynomials: Why do we need to modulo an irreducible polynomial? What happens if this polynomial is reducible? Does such an irreducible ...
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0answers
43 views

A field $K$ of order $q=p^r$ contains a subfield $K'$ of order $q'=p^k$ if and only if $k\mid r$.

I was trying to prove the following theorem: A field $K$ of order $q=p^r$ contains a subfield $K'$ of order $q'=p^k$ if and only if $k\mid r$. My attempt at the proof was: $K'\setminus\{0\}$ is ...
1
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0answers
15 views

How to find an irreducible polynomial over a finite field with a primitive root (and low hamming weight)

I found there that a polynomial in $F[x]$ with $|F| =q $ with degree $n$ will have its roots in $K$ of order $q^n$ Here, I found that either all the roots are primitive or none of them are. I am ...
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0answers
27 views

Does desingularization of projective curve over finite field add new points?

Let $C: F(x,y,z)=0$ be the projective curve over $\mathbb{F}_{13}$ given bellow. $C$ has only two rational points, both singular and the genus of $C$ is one. To satisfy the Hasse-Weil bound, the ...
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1answer
36 views

Factorization of polynomials in Galois Field 2

I want to know what is the general rule for factorization of the generator polynomial which is based on Galois Field, GF2? Fact is, a generator of degree m must divide ...

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