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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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Irreducibility of $t ^ { 4 } + t - 1$ over $\mathbb{F}_{25}$

Problem: Let $f ( t ) = t ^ { 4 } + t - 1 \in \mathbb { F } _ { 5 } [ t ]$, show that $f(t)$ has no roots in $\mathbb{F}_{25}$. I tried the following: suppose $f(t)$ has a root in $\mathbb{F}_{25}$, ...
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0answers
24 views

Must Additive Inverse elements be opposite in sign?

Is it possible to define a field $F=\left \{ 0, 1, a \right \}$ where the Additive Inverse condition is expressed as : $x+x=0 \space \space \forall x\in F$ ? My doubt comes from reading my book on ...
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2answers
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Finding short vectors in $\mbox{GF}(2)$

Suppose we have a system of $n$ linear equations in $m$ unknowns over $\mbox{GF}(2)$ (binary field) $$Av=b$$ Let $V = \mbox{GF}(2)^m$ be the vector space of possible assignments of the variables. For $...
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2answers
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How many generator matrices does an [n, k]q - linear code have?

So far, I have realised that there exists a unique generator matrix for a $[n,k]_q$-linear code if and only if $n=k=1$ and $q=2$. I also believe that the number of generator matrices is given by the ...
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1answer
25 views

If $x\in\mathbb{F}_{q^2}$, then $x+x^q\in\mathbb{F}_{q}$.

I'm reading a paper right now that uses this property but they don't really explain it well. They just write that "it is clear", but it isn't exactly clear to me. Their proof involves choosing a ...
2
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1answer
44 views

Irreducibily of polynomials in two variables

Let $\mathbb{F}_q$ be a finite field where $q$ is odd. Let $f \in \mathbb{F}_q[x,y]$ be the following polynomial $$f:=(x^2y^2 - 2x^2y - 2xy^2 - x^2 + xy - y^2 - 2x - 2y + 1).$$ How to prove that $f$ ...
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3answers
67 views

Find the basis of GF(2)

I do past papers and I stumbled upon this question in one of the papers. I know what is GF(2), but I have no idea how to find the basis from the given data. \begin{bmatrix}1\\1\\0\\0\end{bmatrix} \...
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1answer
49 views

Find a sequence of whole numbers $n _ { 1 } , n _ { 2 } , \ldots$ such that $n _ { i - 1 } | n _ { i }$ for all $i \geq 2$

Problem : Find a sequence of whole numbers $n _ { 1 } , n _ { 2 } , \ldots$ such that $n _ { i - 1 } | n _ { i }$ for all $i \geq 2$ and for every $k \in \mathbb{N}$ there exists $i$ such that $k | n ...
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2answers
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Create projective plane

Please explain this https://math.stackexchange.com/a/463369/672948 in a simpler way. I am not from higher mathematics background and these terms are quite hard to understand. I am clear upto finding ...
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0answers
55 views

Choosing a zero-sum sequence from a finite field. [duplicate]

Let $\mathbb{F}_p$ be a finite field (for a prime integer $p$), and let $x_1, \ldots, x_{2p-1}$ be any sequence of elements from it. Prove that I can chose $p$ elements from this sequence such as ...
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1answer
67 views

Proving an isomorphism of Galois group

Let $p$ be a prime number and $\alpha\in\mathbb{N}$ such that $\forall\beta\in\mathbb{Q} \space\alpha\neq \beta^p $ e.g. $\alpha$ is not a $p$-th power of any rational number. Let $E$ denote the ...
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1answer
49 views

Elliptic Curve over a Finite Field, Adding Graphically

I use Mathematica to add two points graphically on the elliptic curve $y^2 = x^3 + 3x + 8$ over $\mathbb{F}_{13}$. Specifically, I'd like to illustrate $(1,8)+(2,10)=(1,5)=(1,-8)$, but on first glance,...
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2answers
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Subrings $A$ of $\mathbb{F}_p[x]$ such that $\dim_{\mathbb{F}_p}\mathbb{F}_p[x]/A=1$.

Suppose we have the ring $\mathbb{F}_p[x]$ of polynomials with coefficients in the Galois field with $p$ prime number elements. I want to find all subrings $A$ containing the identity such that when ...
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2answers
41 views

Degree of a splitting field of an irreducible polynomial over $\mathbb{F}_p$

Is it true that whenever $p$ is an odd prime, and $f$ an irreducible polynomial of degree $p$ in $\mathbb{F}_p$, then the splitting field of $f$, denoted $L$, satisfies $[L:\mathbb{F}_p] = p!$ ? I ...
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0answers
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Multivariate variate polynomials over finite field with common root will have another root, probabilistic proof?

I have a finite field $F_p$ and $m$ polynomials $P_1, P_2, \dots, P_m$ in this field. Also every polynomial is multivariable, so $P_i$ from $F_p^n$ to $F_p.$ It is known that $n>\sum_i \deg(P_i).$ ...
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1answer
45 views

how many points belong to the quadric $x_0^2+x_1^2+x_2^2+x_3^2=0$ in $\mathbb{P}_3$ over $\mathbb{F}_9$

I have a problem with the following question: how many points belong to the quadric $x_0^2+x_1^2+x_2^2+x_3^2=0$ in $\mathbb{P}_3$ over $\mathbb{F}_9$. How I tried to solve this problem. Here we have ...
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1answer
41 views

If $K$ is an extension of $\mathbb{Z}/p\mathbb{Z}$, every elements of $K$ is a root of $t^{p^n -1}-1$

Problem : Show that the characteristic of a finite field $K$ is a prime number. Show that $K$ is an extension of $\mathbb{Z}/p\mathbb{Z}$ and there exists $n$ such that Card($K$) $= p^n$. ...
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1answer
82 views

Conjugacy of Singer cyclic groups in $\mathrm{P\Gamma L}$

Motivation This is kind of a follow-up to this question on conjugacy of Singer cyclic groups in GL. The "original" definition of a Singer cycle is not in the GL, but the following slightly different ...
2
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1answer
83 views

trace of Frobenius

how can I calculate trace of Frobenius for a single point on an elliptic curve $E(F_{q^{12}})$? I've tried to sum up 12 points that were different powers Frobenius maps but none of the points don't ...
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1answer
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additive order of any zero divisor in $Z_{p^2}$ is p, is it true?

This result was used in a proof of a theorem, i am not sure if it's true. can someone tell the proof idea. Can it be generalized to additive order of any zero divisor in $Z_{p^k}$, is there any ...
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4answers
64 views

Let $F$ be a finite field with $\text{char}(F) = p$. Now, if $u$ is a primitive element, show that $u^p$ is also primitive.

I need help in understanding how to prove this. I know that if $u$ is a primitive element of a Finite field, $F$, then $u$ generates $F^*$.
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1answer
106 views

Interpolation of a rational function

Assume I am given two polynomials $f(x)$ and $g(x)$ with coefficients from a field $\mathbb{F}_p$, where $p$ is a prime. Now I know that the set of these polynomials is a ring and not a field, meaning ...
4
votes
2answers
69 views

Minimal polynomial of extension of degree 2 over a finite field with characteristic 2

I'm struggling to solve the following question. Let $F$ be a finite field with characteristic 2 and $L/F$ be a finite extension with $[L:F]=2$. Prove that there exists $\alpha\in L$ such that $L = ...
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1answer
32 views

A question about subgroup lattice of $GL(2,3)$ [closed]

If I am given a collection of matrices, whose entries are from $\mathbb{Z}_3$, and they are all $2\times 2$, is every subgroup of this group cyclic? And what would a subgroup lattice look like for ...
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3answers
90 views

factorization of polymonials in $\mathbb{Z}/11\mathbb{Z}$

how can we find factorization of polynomials in $\mathbb{Z}/11\mathbb{Z}$? thank you very much in advance.
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0answers
46 views

Help understanding the question

Q) Let $\zeta = e^{\frac{2\pi i}{11}}$. Find a quadratic equation over $Q$ for $x = \sum_{a\in Q}\zeta^a$, where $Q \subset \mathbb{Z}_{11}^{*}$ is the set of squares in $\mathbb{Z}_{11}^{*}$. Does ...
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2answers
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Galois group of $F_{p^2}$ over $\mathbb{Z}_p$

Q) For any prime $p$, what is the Galois group of $F_{p^2}$ over $\mathbb{Z}_p$, where $F_{p^2}$ is the field with $p^2$ elements. I know how to find Galois group of a particular polynomial over a ...
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1answer
48 views

Finite field $\mathbb{F}_3$ [closed]

Let $a$ be a root of the polynomial $x^2+1$, so that $\mathbb{F}_9=\mathbb{F}_3[a]$. I don't understand this statement...
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1answer
33 views

Isomorphisms between finite fields

I know that all finite fields of the same size are isomorphic to one another. I also know that if a polynomial $f(x)$ is irreducible over $\mathbb{Z}[x]$ and of degree $n$ then $$ \frac{\mathbb{Z}_k[...
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1answer
81 views

This matrix is diagonalizable over algebraic closure of $\Bbb F_p$ iff $(n,p)=1$

Let $\sigma_p:\Bbb F_q \to \Bbb F_q$ be the Frobenius automorphism $\sigma_p(x)=x^p$ where $q=p^n$. Now viewing $V=\Bbb F_q$ as a vector space over $\Bbb F_p$ of dimension $n$. Now how to prove that ...
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1answer
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Isomorphism of fields via the forgetful functor

The following article https://tinyurl.com/yydxzxe3 says that "For the case of fields, given a field F and an isomorphism of sets U(F) → S, there is a unique field whose underlying set is S and which ...
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1answer
67 views

Subgroup of finite field is invariant under Frobenius map

Let $q$ be a prime power and $a\in\Bbb{F}_{q^2}$. Let $m$ be a positive integer dividing $q+1 $ and $H \subset\Bbb{F}_{q^2}^{\times}$ a subgroup of order $m(q-1)$. If $a \in H \cup \{0\}$, why is $a^...
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1answer
238 views

Identity of Polynomials in positive charcteristic

In positive charcteristic $p$, we know that for every field element $x\in\mathbb{F}_{p}$ we get $x^p = x$. Then I think (and I might be wrong, but I don't see how) monomials of the form $t^{p^i}\in\...
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0answers
50 views

Catalan's conjecture in $\mathbb F_p$

Catalan's conjecture (Mihăilescu's theorem) states that the only case of two consecutive integer powers is given by the equation $3^2-2^3=1$. We can easily verify that it is not valid in some prime ...
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0answers
27 views

A confusion on simple extension of a root of irreducible polynomial over a finite field

Let $f(x)$ be an irreducible polynomial over $\Bbb F_p$ with $\deg f(x)=m$. Let $K$ be the splitting field of $f(x)$ over $\Bbb F_p$. Let $u$ be arbitrary root of $f(x)$ in $K$. Then the minimal ...
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1answer
25 views

Order of the splitting field of $f(x)$ with $\deg f(x)=11$ over $\Bbb F_p$?

Given $f(x)\in\Bbb F_5[x]$ with $\deg f(x)=11$ for example. Suppose that $f(x)$ is irreducible. We know that the splitting field of $f(x)$ over $\Bbb F_5$ must exist, and its degree is $\le11!$. ...
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1answer
39 views

Polynomial evaluates to quadratic residue in $p$ cases

This exercise popped up in a chapter on Legendre symbols. Let $p$ be a prime $3$ $(\textrm{mod } 4)$, and $f(x) \in \mathbb{F}_p[x]$ a polynomial of odd degree. Show that the number of solutions $(...
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0answers
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$\mathbb{F}_p({\rm i}) = \mathbb{F}_{p^2}$ iff $p=3\mod 4$

I know the fact that \begin{equation*} \mathbb{F}_{p^2} = \mathbb{F}_p({\rm i}) \simeq \mathbb{F}_p[x]/(x^2+1) \end{equation*} for a unit ${\rm i}$, i.e. ${\rm i}^2=-1$, iff $p=3\ {\rm mod}\ 4$ for a ...
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1answer
63 views

How many solutions of the equation $ax^2 +by^2 = 1$ are there with $(x, y) ∈ \mathbb{F}_{p} ×\mathbb{F}_{p}$ [duplicate]

How many solutions of the equation $ax^2 +by^2 = 1$ are there with $(x, y) ∈ \mathbb{F}_{p} ×\mathbb{F}_{p}$ where $a, b$ are integers whose product is not divisible by $p$? This was a recommended ...
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3answers
35 views

How many elements of have square roots in a field of 13 elements? [duplicate]

Initially, I thought since this field was isomorphic to $({0,...,12})$ , the elements $(0,4,9)$ would have square roots. However, when I checked the solutions, the answer was different. Thank you in ...
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1answer
39 views

proving $x$ is the generator of a cyclic group

Show that $x$ is a generator of $(\mathbb{Z}_3[x]/\langle x^3+2x+1\rangle)^*$. I don't understand part of the solution. $x^3+2x+1$ is irreducible in $\mathbb{Z}_3$. Let $a$ be a zero of $x^3+2x+1$ in ...
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0answers
49 views

Finding the generator of a cyclic group

Q) Show that $x$ or $2x$ is a generator of the cyclic group $(\mathbb{Z}_3[x]/\langle f(x)\rangle)^*$ where $f(x)$ is a cubic irreducible polynomial over $\mathbb{Z}_3$. My attempt: Let $F= \mathbb{Z}...
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1answer
86 views

Efficient calculation of difference sets from finite fields

A while ago I wrote a program to generate, amongst other things, difference sets from finite fields. Generating these sets is rather slow. Is there some theorem or construction I could use to speed it ...
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1answer
40 views

Is there an easy expression for multiplicative inverses in $\mathbb Z_p$?

I know that in arbitrary division rings, one can go about finding inverses Euclidean division. But take $\mathbb Z_{11}$ as a simple example. Is there a "nice" expression which yields the inverses in ...
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2answers
40 views

A question about the degree of an extension field

Consider $f(x) := x^3+2x+2$ and the field $\mathbb{Z_3}$. $f(x)$ is obviously irreducible over $\mathbb{Z_3}$. Let $a$ be a root in an extension field of $\mathbb{Z_3}$, then why is it that $[\mathbb{...
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0answers
6 views

$\mathbb{F}_{p}(t)$ separable over $\mathbb{F}_{p}(f(t)/g(t))$ when $\deg(f), \deg(g) < p$.

Let $f(t), g(t) \in \mathbb{F}_{p}[t]\setminus \{0\}$ where $t$ is an indeterminate, and where $\max\{\deg(f), \deg(g)\} < p$ and $f(t)/g(t) \not\in \mathbb{F}_{p}$. Show that the extension $\...
1
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1answer
21 views

A question about finite field extension of a finite field

Let $K$ be a finite extension field of a finite field $F$. Show that there is an element $a\in K$ s.t. $K = F(a)$. My attempt: $K$ is a finite field and $char(K) = char(F) := p$. I know that for a ...
2
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0answers
44 views

A problem about the field of rational functions over finite field

Let $p$ be a prime number, and let $F_{p}$ be the finite field with $p$ elements. Let $F=F_{p}(t)$ be the field of rational functions over $F_{p}$ . Consider all subfields of $F$ such that $F/C$ is a ...
2
votes
2answers
72 views

Factorization of a polynomial in $\Bbb F_7$

I need to reduce as much as possible the polynomial: $x^6+3x^5+2x^4+6x^3+4x^2+5x+2$ in the finite field $\Bbb F_7$. It has no roots over the field, and I don't think that it is necessary to check ...
4
votes
1answer
56 views

Low-degree polynomial $T\in\mathbb F[x,y]$ with $T(P(z),Q(z))=0$

Given polynomials $P,Q\in\mathbb F[z]$ over a finite field $\mathbb F$, one can find a non-zero polynomial $T\in\mathbb F[x,y]$ such that $T(P(z),Q(z))=0$ for any $z\in\mathbb F$. Is there a way to ...