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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15 views

Complexity of a permutation over finite field $\mathbb{F}_{2^n}$ with a specific cycle structure? [closed]

a little motivation for my questions: I have very naive and limited knowledge of mathematics and I'm trying to understand computation. we can represent a computer by a function $f: \mathbb{F}_{2^n}\...
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32 views

Irreducible polynomial in $\mathbb{Z}_{17}$ [duplicate]

Is the polynomial $x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$ irreducible in $\mathbb{Z}_{17}$? I don't know how to solve it. Thank you for any help!
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20 views

Clubs whose intersections are multiples of six (Oddtown variant)

This is a question about generalizing the famous "Clubs in Oddtown" problem. The original setup is that a town has $n$ people, and $m$ clubs each consisting of a subset of the population. ...
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1answer
45 views

The smallest extension field of $\mathbb{F}_3$ containing all the zeros of $f(x)=x^{11}-1$

Firstly notice that $f(1)=0$ so $f(x)=(x+2)g(x)$ over $\mathbb{F}_3$ for some $g$. Moreover, $\text{ord}_{11}3 = 5$, ie $5$ is the smallest exponent $t$ such that $3^t=1 \mod 11$, so $5$ is the lcm of ...
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6answers
210 views

Giving a 1-hour talk to highschool math club: any topic suggestion?

I've been invited (by my kid) to give a one hour talk to her highschool math club. Last year (right before the pandemic hit) I did two such talks on probability, and they loved it. I'm looking for ...
2
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1answer
52 views

Primitive element of $\mathbb{F}_{16}^{\times}$

I have to find a primitive element of $\mathbb{F}_{16}^{\times}$. I defined $\mathbb{F}_{16} = \frac{\mathbb{F}_{2}[a]}{\langle a^4+a+1 \rangle}$. I tried to prove that $a^3$ and $a^5 \neq 1$. I got $...
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1answer
40 views

On the finite algebra $\Bbb F_q[X]_{<n}$ over $\Bbb F_q$

Let us take the vector space of all the polynomials of degree less than $n$ over the finite field $\Bbb F_q$, \begin{eqnarray*} \Bbb F_q[X]_{<n} &:=& \{ f(X)\in \Bbb F_q[X]:\deg f(X) <n\}...
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The best methods for multivariate polynomial equations over finite fields

I am trying to get a rough overview of the best methods one might use to find solutions of multivariate polynomial equations over large finite fields. We can suppose for simplicity that the given ...
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23 views

Defining set of a cyclic code

I am asked to find a defining set of a binary cyclic code of length 15 which isn't a BCH code. I already found the 2-cyclotomic cosets: $C_0 = \{0\}$, $C_1 = \{1,2,4,8\}$, $C_3 = \{3,6,9,12\}$, $C_5 = ...
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LCM in $\mathbb F_q[T]$

Let $q$ be a power of a prime $p$. We work in $\mathbb F_q[T]$. Put $L_n=\prod_{j=1}^n(T^{q^i}-T)$. Does one have $$\deg(\mathrm{LCM}(L_{n+1};L^q_n))=\frac{q^{n+2}}{q-1}+o(q^n)$$ when $n\to\infty$?
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Show if the following polynomials are reducible or not

I am asked to show if $f(x)=x^3+x+1$ over $\mathbb{F}_5$ $g(x)=x^4+x+1$ over $\mathbb{F}_2$ are reducible or not. I was able to show in each case they are irreducible. For $f(x)$ I assumed it is ...
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120 views

Number of points on the elliptic curve $y^2 = x^3 - x$ over $\mathbb{F}_q$

I need to know the value of $\sum_{x \in \mathbb{F}_q} \chi(x^3 - x)$, where $q = p^r$ with $p$ prime and $\chi$ is the Jacobi symbol. If we take $-x$, we have $\chi((-x)^3 - (-x)) = \chi(-1)\chi(x^3 -...
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1answer
24 views

How many $(a, b) \in \mathbb{Z}_p^2$ are there such that $ax + b = y \mod p$ for some $x, y \in \mathbb{Z}_p$?

Background: I am trying to convince myself that the family of hash functions $H = \{h_{(a, b)} : \mathbb{Z}_p \rightarrow \mathbb{Z_p}, (a, b) \in \mathbb{Z}_p^2\}$, where $h_{(a, b)}(x) = ax + b \mod{...
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1answer
39 views

Number of points of the elliptic curve $y^2 + y = x^3$ over $F_q$

Given the elliptic curve $y^2 + y = x^3$ over $\mathbb{F}_q$ ($q=p^r$, where $p$ is prime), I want to prove that if $q \equiv 2 \bmod 3$, then the elliptic curve has $q + 1$ points. My exercise says ...
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1answer
54 views

Adjoining element in a finite field $\mathbb F_{p^n}$ to $\mathbb F_p$ to get $\mathbb F_{p^6}$

Given that $p$ is a prime and $n$ is a natural number, I want to know how many $\alpha$ are in $\mathbb F_{p^n}$ such that $\mathbb F_p (\alpha) = \mathbb F_{p^6}$. I know that $[\mathbb F_{p^6} : \...
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1answer
32 views

Lower bound for number of points of elliptic curve over finite field

I'm asked this kind of question: Show there exists $q+1$ points on the elliptic curve over $\mathbb{F}_q$ given by $y^2 = x^3-x$ when $q \equiv 3 (mod 4)$ The fact that I'm asked for an ...
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1answer
33 views

How to extend scalars of a MeatAxe $kG$-module $M$ given in GAP?

Let $G$ be a finite group. Suppose $k:=\mathbb{F}_5$, let char$(k)\mid |G|$, and let $M$ be a $kG$-module given in GAP, as in the following example: ...
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1answer
62 views

Extension Galois field, multiplication of elements

Consider the extension field $GF(2^4)$ and the primitive polynomial $P(x)=x^4+x^3+1$. How would I find the result of multiplication of elements $1011$ and $1100$? My work: $1011$ corresponds to $\{1x^...
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1answer
40 views

Prove, that the multiplicative group of a finite field is cyclic. [duplicate]

I have almost completed the proof, I have shown, that there exists an element of order e, where e is min{n:$g^n=1$} and $g\in G$ where $G$ is the multiplicative group of a finite field. Now I noticed, ...
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1answer
34 views

Express a irreducible polynomial $f(X)$ in the field of characteristic $p>0$ $F[X]$ as $g(X^{p^m})$

$F[X]$ is of the form $a_0+a_1X+...+a_nX^n$ where $a_0,a_1,...a_n\in F\text{ with characteristic p>0}.$ Express $f\in F[X]\ as\ g(X^{p^m})$, where the nonnegative integer m is a large as possible ...
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Period of a linear recurrence mod p

$\newcommand{\FF}{\mathbb{F}} $ $\newcommand{\bs}{\mathbf{s}}$ Given linear recurrence $$ x_n = a_{1}x_{n-1} + a_{2}x_{n-2} + \cdots + a_kx_{n-k}\pmod p, $$ for $x_0, x_1, \ldots$ we define its state ...
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1answer
34 views

Confusion on the order of $GL_2(\mathbb{F}_p)$

In this question, they show that the order of $GL_2(\mathbb{F}_p)$ is $(p^2-1)(p^2-p)$. For the first column, there are $p^2$ options, and we need to exclude the $0$ column, so there are $p^2-1$. That'...
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29 views

Polynomials divide each other in a finite field implies they are constant multiples of each other

If two polynomials divide each other in a field such as $\mathbb{R}$ or $\mathbb{Q}$, then they certainly have to differ by a constant: $g(x) = k f(x)$. But what about a finite field? If polynomials ...
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21 views

Existence of an indicator function in finite field notation

The indicator function on a set $A$ is a function $1_A:A\to \{0,1\}$ defined by $$1_A(x)= \begin{cases} 1 &\text{if } x\in A \text{ and}\\ 0 &\text{if } x\notin A. \end{cases}$$ More ...
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Constructing a “microscopic” error correcting code

I'm a doctor studying tiny specimen under a microscope. I had this strange idea to "barcode" each specimen with a physical barcode. This barcode is assembled from two pigments which can be ...
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45 views

$[\mathbb{F}_{p^n}: \mathbb{F}_p] = n$?

I have seen multiple answers using this fact: $[\mathbb{F}_{p^n}: \mathbb{F}_p] = n$. Proving that $f(x)$ divides $x^{p^n} - x$ iff $\deg f(x)$ divides $n$ Prove that if $\mathbb{F}_{p^n} \subseteq \...
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1answer
42 views

Determinant of $\mathbb{F}_2$ square matrices

Let $A_1,A_2,A_3$ and $B$ be in $\mathbb F_2^{n\times n}$. Is $$\mathsf{Det}(A_{1}A_2A_{3}+B)=\mathsf{Det}(A_{1}A_3A_{2}+B)=\mathsf{Det}(A_{3}A_2A_{1}+B)=\mathsf{Det}(A_{2}A_3A_{1}+B)=\mathsf{Det}(A_{...
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1answer
32 views

Number of squares of the form $a^2+x^2$ in a finite field

Suppose $a$ is a non-zero element in a finite field $GF(q)$ of odd characteristic. How many $x \in GF(q)$ are such that $a^2+x^2$ is a square in $GF(q)$? From some experiments with some small fields, ...
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42 views

Let $F$ be a field with $|F|=q$ and $[K:F]=2$. Let $α∈F$ of order $q-1$. Then there exist an element $β \in K$ of order $q^2-1$ such that $β^{q+1}=α$

Let $F$ be a field with $|F|=q$ and $[K:F]=2$. Let $α∈F$ of order $q-1$. Then there exist an element $β \in K$ of order $q^2-1$ such that $β^{q+1}=α$ Am stuck with finding such a $\beta$ of order $q^...
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1answer
64 views

isomorphism between $\frac{\mathbb{F}_5[x]}{(x^2+x+1)} $ and $ \frac{\mathbb{F}_5[x]}{(x^2 -2)} $

I know $\frac{\mathbb{F}_5[x]}{(x^2+x+1)} $ and $ \frac{\mathbb{F}_5[x]}{(x^2 -2)} $ is isomorphic because they are both 2-degree extension of $ \mathbb{F}_5 $ . But I cannot contract explicit ...
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0answers
31 views

Cayley table for the additive and mutiplicative operations in ($\Bbb F_7$, +, *)

I am to construct the Cayley tables for the additive and multiplicative operation in $(\Bbb F_7, +, *)$. I have started by stating The order (nr of elements) of a finite field must be a prime or ...
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0answers
58 views

Linear-algebraic interpretation of $q$-multichoose

The $q$-analog $[n]$ of a whole number $n$ is $q^{n-1}+\cdots+q+1$, in which case the binomial and $q$-binomial are $$ \binom{n}{k}=\frac{n(n-1)\cdots}{k(k-1)\cdots} \qquad \left[\begin{matrix} n \\ k ...
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3answers
60 views

Show that a finite domain is a division ring [duplicate]

Let $R$ be a finite ring. Show that the following are equivalent: i. $R$ is a division ring. ii. $R$ is nontrivial and if $r$,$s \in R$, with $rs=0$, then either $r=0$ or $s=0$. $\textbf{NOTE:}$ A ...
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1answer
44 views

What motivation for assuming field K is infinite.

A theorem in Shaum's Outline of Linear Algebra says that if we suppose the field $K$ is infinite then any system of linear equations has either a unique solution, or no solution, or an infinite number ...
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1answer
34 views

Contradiction with Nth roots on finite fields

I am having problems finding the mistake in my thought process here. One of the results from Ireland-Rosen on the topic: If $F$ is a finite field with $q$ elements then for every $\alpha \in F^*$ , $...
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1answer
48 views

Prime of the form $n^2-2m^2$

I need to prove that a prime $p$ is of the form $n^2-2m^2$ iff $p=2$ or $p=8k\pm 1$. So first I tried figuring out, for which $p$ does $n^2-2m^2=0$ has a solution if $\mathbb{F}_p$, or similarly all ...
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1answer
39 views

Squares in a finite field $\mathbb{F}_p$

I need to find all prime $p$ s.t $n+3$ is the inverse of $n-3$ in $\mathbb{F}_p$. So obviously this means $(n+3)(n-3)=1\mod p$, meaning $n^2=10\mod p$. So the question is - for which $p$ does the ...
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1answer
73 views

Definition of Frobenius automorphism

I quote the following problem from the chapter Hilbert's Ramification Theory of Jurgen Neukirch Let $L/K$ is a Galois extension with prime ideal $\mathfrak{P}$, unramified over $K$, then there is one ...
4
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1answer
90 views

An isomorphism between two finite fields

Suppose we have two fields $F_1$ and $F_2$ of order $9$ where both groups of units are cyclic, i.e. $$F_1=\{0\}\cup\{\alpha^i\,|\,0\leq i\leq 7\},\qquad F_2=\{0\}\cup\{\beta^i\,|\,0\leq i\leq 7\}$$ ...
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1answer
136 views

The splitting field of polynomial over $Z/2Z$

I am trying to figure out the splitting field of $x^4 + x^3 + 1$ over $\mathbb{F}_2$ . I know $x^4 + x^3 + 1$ over $\mathbb{F}_2$ is irreducible. Let $α$ be a root of $x^4 + x^3 + 1$, then the ...
2
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1answer
53 views

multiply and sum in Finite Field

$F_q$ is a finite field of $q$ (which equals $p^n$,p is a prime number). $F_q^*$ denotes the elements in $F_q$ which has inverse. Prove that: The multiply of all the elements in $F_q^*$ equals -1. $p>...
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35 views

If $f(x)=ax^{2p}+bx^p+c\in\mathbb{F}_p[x]$, prove that $f'(x)=0$

If $R$ is a commutative ring, then the set of all polynomials with coefficients in $R$ is denoted by $R[x]$. $\mathbb{I}_p[m]$ is the integers mod $m$. When $p$ is a prime, we will usually denote the ...
3
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1answer
27 views

Union of all finite fields of order $p^i$, for some prime number $p$ is algebraically closed. [duplicate]

Let $p$ be a prime number. Consider for all $k,r\in\mathbb{N}_0$ with $k|r$ the field $\mathbb{F}_{p^k}$ as a subfield of $\mathbb{F}_{p^r}$. Define $$ \overline{\mathbb{F}_p}:=\bigcup_{i\in\mathbb{N}...
2
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0answers
33 views

When is a two-dimensional representation of a finite group over a finite field defined over a subfield?

I'm reading the proof of the Deligne-Serre theorem attaching Galois representations to newforms of weight one, and there's a representation-theoretic argument that I don't understand at all. The setup ...
2
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1answer
22 views

Prove or confute the follow proposition

I have to prove or confute the follow proposition: Let $F$ a field and $ f: $$\mathbb Z \rightarrow F$ an homomorphism of rings such that $f(1_\mathbb Z ) =1_F$. If $f$ isn't injective $\Rightarrow$ $...
2
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0answers
40 views

Galois covers of curves of arbitrary degree.

For smooth projective algebraic curves over finite fields. Do they admit a finite Galois cover of any degree, that is not induced by just base extension?
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3answers
39 views

How to calculate with $\mathbb{Z}/2\mathbb{Z}$ with an unknown variable?

When calculating with numbers from a $\mathbb{Z}/2\mathbb{Z}$ how do you deal with unknown variables? For example, if I have the following term: $(a - 1)(a - 1) - (a - 1) - (a - 1) = a^2 + 1$ Or is ...
3
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1answer
57 views

Find the field that $\mathbb{Z}_7[x,y]/\langle y-2x^2, 4xy + y +1 \rangle$ is isomorphic to

See the title. Here the set $\mathbb{Z}_7 = \{0,1,2,3,4,5,6\}$ is considered a ring with the obvious operations $+$ and $\cdot$; and $I := \langle y-2x^2, 4xy + y + 1 \rangle$ is an ideal in $\mathbb{...
0
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0answers
56 views

How many morphism $\mathbb{F}_q\to \mathbb{F}_{q^n}$

Sorry for my bad English. Let $q$ be prime $p$ power, and $n>0$ be integer, and $\mathbb{F}_q$, $\mathbb{F}_{q^n}$ be finite fields. Now how many morphisms of field $\mathbb{F}_q\to \mathbb{F}_{q^n}...
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1answer
31 views

If $f$ is not injective then $F$ is finite.

I have to prove or confute the follow proposition: Let $F$ a Field and $f:\mathbb{Z} \to F$ an homomorphism of rings such that $f(1_\mathbb{Z})=1_F$. Show that if $f$ isn't injective then $F$ is ...

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