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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On the complexity of global fields isomorphism

Let $L$ and $K$ be isomorphic global fields (i.e. either function fields of curves over a finite field, or number fields). What is the complexity of finding an isomorphism between them? Is there a ...
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$\mathbb{F}_{p}[X]/(X^{2} + X + 1)$ is a field if and only if $p \equiv 2 \mod 3$

I have seen this question in some other posts, but I still have some concerns regarding the proof. Question: Let $p$ be a prime number. Show that $\mathbb{F}_{p}[X]/(X^{2} + X + 1)$ is a field if and ...
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Minimal polynomial of all elements in a finite field extension of $\mathbb{F}_3$.

Let $i \in \mathbb{F}_3$ be a zero of $x^2+ 1$. Let $\mathbb{F} = \mathbb{F}_3(i)$, and determine the minimal polynomial of $\alpha\in \mathbb{F}$ over $\mathbb{F}_3$. I'm trying to solve the question ...
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$f = X^{3} + 2$ irreducible in $\mathbb{F}_{49}[X]$.

Question: Prove that $f = X^{3} + 2$ is irreducible in $\mathbb{F}_{49}[X]$. Is $f$ irreducible in $\mathbb{F}_{7^{n}}$ for all even $n$? My attempt: Notice that $\deg(f) = 3$, therefore $f$ is ...
ByteBlitzer's user avatar
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Connected components of subgroup of torus

Consider a finite field $\mathbb{F}_q$ of characteristic $p>0$. Let $A=(a_{ij})$ be an integer matrix with $k$ columns and a finite number of rows. Consider the algebraic subgroup $\pmb{H}_A$ of ...
abeli's user avatar
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Action of $SL_2(\mathbb{Z})$ on the projective plane over $\mathbb{Z}_p$

The group $SL_2(\mathbb{Z})$ act on the projective spaces $P(\mathbb{Z}_p)$ and the upper half of the complex plane $\mathbb{H}$ by linear fractional transformations. I am wondering whether there is a ...
QMath's user avatar
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Why is $\langle 3 \rangle$ a generator of $(\mathbb{Z}_7^*, \cdot)$?

This link here shows that $\langle 3 \rangle$ is a generator for the given group by brute force method, that is trial and error. I was curious as to how to justify this using the theorem. According to ...
The Wanderer's user avatar
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Reference for: Every element of a finite field is the sum of two squares.

Note: This is a reference-request question. It doesn't require the usual level of context. I am not asking for a proof. It is well known that Theorem: Let $x\in F$ for a finite field $F$. Then there ...
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Question over monic polynomials

let $ p $ be a prime number. Consider $ \Bbb Z_p[x] $. Is possible that for a certain polynomial degree bigger or equal to one ; the only irriducibile polynomials of that degree were non monic?
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Under $ad-bc=1$, is every element of a finite field of the form $a^2+b^2+c^2+d^2$?

The Question: Let $x\in\Bbb F_q$, where $\Bbb F_q$ is the field of $q=p^r$ elements, $p$ prime, $r\in\Bbb N$. Can we write $$x=a^2+b^2+c^2+d^2\tag{1}$$ for $a,b,c,d\in\Bbb F_q$ such that $ad-bc=1$? ...
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Construction of Type II optimal normal basis of $GF(2^n)$ over $GF(2)$

Substituting value $n=2$ in Construction theorem of Type II Optimal normal basis For $n=2$, ONB of Type II exist for $\mathbb{F}_{2^n}$ over $\mathbb{F}_2$. because, 2 is primitive $\mathbb{Z}_5$. By ...
Akhilesh Ajithan's user avatar
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What are $0$ and $1$, the elements of $\mathbb Z/2\mathbb Z$? How do they relate to $\mathbb Z$? [duplicate]

I often see field $\mathbf{Z}/\mathbf{2Z}=\{0,1\}$. Without other indication we might see elements of field $\mathbf{Z}/\mathbf{2Z}$ as a subset of $\mathbf{Z}$. Operations such as $2+1=3=1$ are also ...
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When is the restriction of this bilinear form non-degenerate?

In Jürgen Elstrodt's Measure and Integration Theory (8th German edition) there is the following exercise I.3.6: Exercise: The group $(\mathfrak{P}(X), \Delta)$ (power set with symmetric difference) is ...
Zufallskonstante's user avatar
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Do Singer cycles create all matrices of maximal order in $\operatorname{GL}_n(\mathbb{F}_q)$?

I am interested in elements of $\operatorname{GL}_n(\mathbb{F}_q)$ of maximal order. It is known that the maximum possible order of such an element is $q^n-1$ and that this bound is achievable via ...
Randall's user avatar
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How to define the multiplicative group on the additive group of a finite field?

For a finite field $\mathbb{F}_{p^n}$ with characteristic $p$ , we can with the Fundamental Theorem of Finitely Generated Abelian Groups and the elementary divisor decomposition to show that the ...
Dian Wei's user avatar
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A self orthogonal additive linear code over GF(4) is even because $u*(wu) = wt(u)mod2$

Question drawn from Calderbank. et. al's paper "Quantum Error Correction via Codes over $GF(4)$". All codes are assumed to be additive and are over $GF(4)$. It refers to theorem $4$: "...
am567's user avatar
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Consistent Choice of Root of Unity in Characteristic $p > 0$

Let $\zeta_n$ be a primitive root of $x^n - 1$ in $\bar{\mathbb{F}}_p$ where $p \nmid n$. I'm assuming there's no canonical choice for $\zeta_n$ like in characteristic $0$, where we can just take $e^{...
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For elliptic curves, how to express the condition at which the vertical tangent line L, to curve E, at point P is at infinity point?

So I'm writing an essay for my school program. I'm trying to convey the two cases at which P + Q = (a point at infinity). I've written the following for case 1: (1) P != Q and xp = xq then P + Q = (...
Juan The IB Student's user avatar
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Some examples of the following irreducibility test and its name.

Here is a test I have taken in my algebra class: If $f \in \mathbb Z[X]$ and a prime $p$ does not divide the leading coefficient and $\bar{f} \in \mathbb F_p[X]$ is irreducible, then $f$ is ...
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Finite extension of finite field

I have two questions: Suppose $F$ is a finite field then prove that it is a simple extension of its prime subfield. Suppose $F$ is a finite extension of a finite field $K$. Then prove that $F$ is a ...
RIPAN DAS's user avatar
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Reference for Cardinality of Parabolic Subgroup of Symplectic Group over Finite Fields

I am looking for a source to reference that gives the cardinality of parabolic subgroups of the Symplectic group $Sp$ over a finite field $\mathbb F_q$. What I want is essentially exactly what is in ...
Ryan L's user avatar
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Finite field with infinitesimals / nonstandard analysis over finite fields

I have two questions, which are really the same question phrased in two ways: Has there been any research on adjoining infinitesimal elements to finite fields? Has anyone considered extensions to ...
Jim's user avatar
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definition of the operations and additive identity for fields

I know that if the ordered triple $(S,+,*)$ is a ring , then the sign $+$ and $*$ can represent some operations different from their original usage such as addition and multiplication(for example ...
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1 answer
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Irreducible polynomials in $\mathbb F_q[T]$

Let $q$ be a power of a prime $p$. Is there an infinite set $S$ of $\mathbb N$ such that for every $l\in S$, the polynomial $T^{q^l}-T-1$ is irreducible in $\mathbb F_q[T]$. It looks like Artin-...
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Need example of a finite noncommutative ring with inverses

I'm trying to write some test cases for set of code, and I need an example of a finite ring (with identity) which is not commutative, but has inverses (for non-zero elements) and whose additive ...
Jim Newton's user avatar
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The multiplicative group of a finite field is cyclic, using modules

I need to prove, specifically using modules over $\Bbb{Z}$, that the multiplicative group of a finite field is cyclic. This is what I've done already: The multiplicative group of a finite field is in ...
soggycornflakes's user avatar
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2 answers
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Reed-Solomon Code RS16(17, 19) - Number of Correctable and Detectable Symbols

In Reed-Solomon coding with 16-bit symbols, for a configuration RS16(17, 19) where there are 17 data symbols, 2 parity symbols, and a total of 19 symbols, how many symbols are correctable and how many ...
cashew_nuts's user avatar
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Factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$

How do I factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$? I checked that the discriminant $D = 16 -4 = 12$ is not a square ($12^{14} = -1 \mod 29$) so this polynomial has no roots. Therefore it's ...
Invincible's user avatar
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Coincidence of numbers of solutions of matrix equations $A^2=B^2$ and $AB=0$ of size 2 over finite fields

Let ring $R:=M_2(\mathbb F_p)$, where prime $p\ne 2$. There are two interesting equations $A^2=B^2$ and $(A+B)(A-B)=0$, and the numbers of solution pairs $(A,B)\in R^2$ of the equations are denoted $...
cybcat's user avatar
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For what values of $q$ is the action of $SL_2(\mathbb{F}_q)$ on $\mathbb{P}^1(\mathbb{F}_q)$ alternating?

The group $SL_2(\mathbb{F}_q)$ acts on the projective line $\mathbb{P}^1(\mathbb{F}_q)$ (faithfully if $q$ is a power of 2, otherwise with kernel $\{\pm I\}$). We say this action is alternating if the ...
hunter's user avatar
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Why does a polynomial which shares a factor with $x^5 - x$ modulo 3 necessarily also share a factor with $-1$? [duplicate]

My question regards a very specific part of an example in Chapter 14.8 of Dummit and Foote (page 641, image shown below) in which the authors are computing the Galois group of $f(x) = x^5 - x - 1$ ...
LéKitty's user avatar
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irreducible curves over $F_q$ with arbitrary number of rational point

There is an exercise in these notes of Pete L. Clark that says that for every $N$ there exist a "nice" curve over $F_q$ that has more than $N$ points over $F_q$(q is fixed). Of course one ...
ALi1373's user avatar
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About the number of rational points of a curve over finite fields

Let $n\geq 1$ be an integer and $b \in \mathbb{F}_{3^{n}}$ be an element such that $b^{\frac{3^{n}-1}{2}}=(-1)^{n+1}$ (we fix $n$ and $b$). Let $C$ be the curve defined over $\mathbb{F}_{3^{n}}$ by ...
rm0329's user avatar
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1 answer
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Quotient Ring of $\mathbb{Z}_3\lbrack x \rbrack$

Find all values of $b$ in $\mathbb{Z}_3$ such that the quotient ring, \begin{align*} R:= \frac{\mathbb{Z}_3\lbrack x \rbrack}{(x^3+x^2+bx+1)}, \end{align*}is a field. I would appreciate a double check ...
Important_man74's user avatar
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$\mid 1 + w + w \mid = \mid 1 + w^2 + w^2 \mid$ where $w=e_p(1)$

I was trying to digest something related to exponential sum however there was obstacle for me. My question is the following: $\mid \sum _{i=1} ^3 e_p(a_i) \mid$ where $e_p (a_i)=e^{\frac{2\pi.i}{p}a_i}...
Fuat Ray's user avatar
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Vandermonde hyperplanes: affine general position?

let $K$ be a field (e.g., a finite field). Fix a dimension $n$, and, for $m > n$, $m$ distinct elements $v_1, \ldots , v_m$ of $K$. consider the following Vandermonde-like matrix: \begin{equation} \...
BD107's user avatar
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1 answer
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A finite division ring $D$ is a field

7.8.12 Wedderburn: A finite division ring $D$ is a field. I have understood several theorems from this book (A first course in abstract algebra by Hiram, paley) by my own, but this one is beyond ...
N00BMaster's user avatar
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Exists abelian extension of $\mathbb{Q}$ with index $p^{n}$ and $K=F[\alpha]$, $f \in F[x]$ the minimal of $\alpha,f(\beta)=0$,so $\beta=\alpha^{p^k}$

I have 2-problems for help: Given a prime $p$ and a positive integer $n$: Problem (1): Show that there is an Abelian extension [i.e., Galois with Abelian Galois group] $K$ of $\mathbb{Q}$ with $[K:\...
TrItOs's user avatar
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10 votes
1 answer
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$3\times 3$ matrices over $GF(3)$ which satisfy $A^{-1}=(A^T)^2$

I came across this question on Quora: How many $3\times 3$ matrices over the finite field $\{0,1,2\}$ satisfy the condition $A^{-1}=(A^T)^2$? Thanks to NumPy, with sheer brute-force, I found that ...
Nothing special's user avatar
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Irreducible polynomials in the field with 4 elements

Let $\mathbb F$ be a field with four elements. Find irreducible polynomials over $\mathbb F$ of degrees $2,3$ and $4$. I got this M.SE answer, which seems very interesting and fit general cases also. ...
N00BMaster's user avatar
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P-adic numbers and quotients

I'm currently reading the Elliptic Curves Number Theory and Cryptography. There is a proof on page 164. As far as I understand, it uses p-adic notation. I'm not very confident with the p-adics yet, so ...
SarkoxedaF's user avatar
4 votes
2 answers
276 views

Evaluate binary determinant

Let $\alpha=(a_1, \dots, a_n), \beta=(b_1, \dots, b_n) \in \mathbb{F}_2^n$. Let $E$ denote the identity matrix. I am trying to evaluate determinant $\det(E + \alpha^T\beta) $. Some experiments show ...
Orel_Algebraist's user avatar
1 vote
3 answers
81 views

Show that $\mid \sum _{i=1} ^n e^{\frac{2\pi.i}{p}a_i} \mid \ge n. \cos(\frac{2\pi}{p})$

I am currently struggling with exponential sum for finite fields and here is the question. Let $p$ be prime and $a_1, \dots,a_n \in \mathbb F_p$ such that $a_i \in [-1,1]$. Show that $\mid \sum _{i=1}...
Fuat Ray's user avatar
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4 votes
2 answers
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Representing finite fields

I was reading Field Theory. Few basic things I know are- For every prime $p$ and natural number $n$, there exist a finite field of order $p^n$. Multiplicative group of finite field is cyclic. So, if ...
Derwal Meena's user avatar
3 votes
2 answers
150 views

Understanding a proof of the Wedderburn theorem

Wedderburn theorem: Every finite division ring is a field. The following proof seem easier than other which I manage to understand the theorem. Although some logic(highlighted ones) still seems ...
N00BMaster's user avatar
1 vote
1 answer
112 views

Conclude: $-3 \in K$ is a square if and only if $q \equiv 1 \bmod 3$. [duplicate]

Let $p \neq 2,3$ be a prime number, $q=p^n$ for an $n \in \mathbb{N}$ and $K$ a field with $q$ elements. Prove the following: $-3 \in K$ is a square if and only if $q \equiv 1 \bmod 3$. My idea: We ...
MathJason's user avatar
1 vote
1 answer
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Eisenstein-irreducibility proof check

I have a polynomial $f(x) =x^6+2x^3-1$ in $\Bbb Q[x]$ I want to check that if this is reducible over $\Bbb Q[x]$. There is classical and long way of checking that by assuming it is factorizable and ...
sknasmd's user avatar
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Unexpected Result from Finite Field Calculations in GF(2^8)

I'm performing calculations within the finite field $GF(2^8)$ and I can't seem to get the expected results. This is my first time working with finite fields, so my understanding is quite basic. I ...
DurangoOlsen's user avatar
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Finite field as $F_p$ vector space

I read GTM 167 Field and Galois Theory written by Morandi.In charpter 2 ,finite fields part, author claims that a finite field $\rm F$whose $\rm{Char(F)=p}$,$\rm p$ is a prime here,can be considered ...
Gary Ng's user avatar
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1 vote
1 answer
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Distinct derivations of polynomial over finite field

I am a student studying algebra and cryptography. I wonder below question is possible. Can I make some polynomials $f(x)$ over finite field that all derivations $f^{(k)}(x)$ are distinct when x is ...
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