Finite fields are structures arising in abstract algebra. 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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root of binary matrix

There is a square matrix A defined over the field GF(2). It means there are zeros and ones in its cells, xor stands for element summation, logical and - for multiplication. Is there any way to find ...
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Quotient Space over finite field.

I'm looking at a vector space $V = F^3$ where $F = \{0,1,2\}$ the field with three elements. I have a subspace $W = \text{span}(1,2,1)$ and I'm trying to explicitly describe the quotient space $V/W$. ...
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Approximating a Euclidean Algorithm

Given the problem of computing the GCD of two given elements over any finite field with characteristic 2. $$ r_1 = q_1r_2 + r_3 \\ r_2 = q_2r_3 + r_4 \\ r_3 = q_3r_4 + r_5 \\ \vdots \\ ...
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204 views

Finite factor ring

I'm reading the paper "How to use finite fields for problems concerning infinite fields" of Jean-Pierre Serre. In pp. 2, Serre uses the fact that, if $\Lambda\subset\mathbb C$ is a ring finitely ...
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315 views

Irreducible Polynomial of degree N

In my project I need to generate addition and multiplication table of $GF(2^N)$. I think first of all i need irreducible Polynomial of degree $N$. So, Is there any algorithm to find Irreducible ...
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279 views

Determine if a polynomial on a finite field is separable

If I want to determine if a given polynomial $P$ over a finite field $\mathbb{F}_q$ is separable, what are the possibilities ? I mean : What is the general method ? I think it's to compute the GCD ...
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Deligne and the four Weil statements about polynomials over finite fields?

This article mentions he recently won 3 million of some money. Does anyone have a link to the original Deligne paper that proves this, and could anyone give a basic rundown of how "counting points on ...
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51 views

How do I add the elements of a finite field knowing the multiplicative structure?

Let $F$ be a finite field then the multiplicative part $F^\times$ is a cyclic group generated by $f$. What - when nonzero - is $f^i + f^j$ as a power of $f$? What is 1,2,3,4,.. in terms of $f$? ...
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Proof that $t^8+2t^6+4t^4+t^2+1$ is reducible in $\mathbb{F}_p$

Prove that the polynomial $t^8+2t^6+4t^4+t^2+1$ is reducible in $\Bbb F_p$, for all $p\in \Bbb P$. Here are some examples: $t^8+2t^6+4t^4+t^2+1=(1 + t + t^4)^2\pmod{2}$ $t^8+2t^6+4t^4+t^2+1=(1 + t) (2 ...
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“Randomize” output of a Linear Feedback Shift Register for the same taps?

I'm using a (Galois) LFSR to sample a large array, ensuring that each entry is only visited once. I simply skip past the entries that exceed the array length. With the same taps then the array entry ...
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289 views

Sylow $p$-subgroups of Finite Matrix Groups

Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, with $n\le p$, and let $P$ be a Sylow $p$-subgroup of $G$. Do all non-trivial elements of $P$ have order $p$? I believe the answer is yes, because I ...
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213 views

How does trigonometry in a Galois field work?

This is a follow-up to this question. I'm interested in doing trigonometry in finite fields on a computer. I do not understand precisely how trigonometric functions are supposed to work in a finite ...
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953 views

Monic Irreducible Polynomials over Finite Field

Let $F=\mathbb{F}_{q}$ be a finite field (so $q=p^k$ for some prime $p$ and positive integer $k$), and let $\varphi(d)$ denote the number of monic irreducible polynomials of degree $d$ in $F[X]$. I'm ...
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879 views

Why does rational trigonometry not work over a field of Characteristic 2?

I am interested in solving triangles in a finite field with a computer program. Rational trigonometry seems well suited to do this. However, the Wikipedia article, as well as several published ...
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166 views

Factoring $x^8-x^4+1$ over $GF(7)$

Could anyone suggest any good way to do it? (The only way I can think of is by looking for roots (There are none), checking a factorization into the product of a 6 and a 2 polynomial (Many unknowns ...
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112 views

irreducibilty of a polynomial over finite field

$f=x^4-x^3+14x^2+5x+16$, considering it a polynomial with coefficient in $\mathbb{F}_3$, it has no roots Considering it a polynomial with coefficient in $\mathbb{F}_3$,it is a product of two ...
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200 views

Translation of a Paper of Tate

I'm just wondering if anyone knows if the paper "Classes d'isogenie des varietes abeliennes sur un corps fini" by John Tate has been translated into English. My French is not that good and I found it ...
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238 views

$E/\mathbb F_q$ extension field. Show $[\mathbb F_q(\alpha) : \mathbb F_q]$ is smallest $n$ satisfying property.

Let $q = p^m$. Suppose that $E/\mathbb F_q$ is an extension field and $\alpha \in E$ is algebraic over $\mathbb F_q$. Show that $[\mathbb F_q(\alpha) : \mathbb F_q] = $ the smallest positive integer ...
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50 views

Representing an element mod $n$ as a product of two primes

Given a positive integer $n$ and $x \in (\mathbb{Z}/n\mathbb{Z})^*$ what is the most efficient way to find primes $q_1,q_2$ st $$q_1q_2 \equiv x \bmod n$$ when $n$ is large? One option is just to ...
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93 views

Prove that in $\mathbb{Z}_p$, $[a]^{-1} = [a]$ if and only if $[a]=[1]$ or $[a]=[p-1]$

Let $p$ be a prime and a an integer. Prove that in $\mathbb{Z}_p$, $[a]^{-1} = [a]$ if and only if $[a]=[1]$ or $[a]=[p-1]$. I greatly appreciate your help on this question!
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Subgroup structure of $\mathrm{SL}(2, p^2)$ and its irreducible characters

I am taking a course in representation theory of finite groups, and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and ...
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151 views

On a characterization of primitive polynomials over a finite field

Let $K$ be a finite field. Let us define a primitive polynomial as an $f \in K[X]$ s.t. the multiplicative order of $X$ in $K[X]/(f)$ is equal to $|K|^{\deg f} - 1$. I want to show that $f \in K[X]$ ...
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195 views

finite field to rational fraction

Suppose I have a number $n\in\mathbb F_p$, i.e. an element of the finite field obtained by arithmetic modulo some (odd) prime $p$. I'm looking for a way to find a simple description of $n$ as a ...
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198 views

Orientation on finite dimensional vector spaces over finite fields.

For finite-dimensional $\mathbb R$-vector spaces, we define an orientation to be an equivalence class of ordered bases, where $B_1 \sim B_2$ iff the change of basis matrix $A$ taking $B_{1}$ to ...
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415 views

Using Hensel's Lemma to Factor a Polynomial over $\mathbb{Z}_4[x]$

We recently learned about codes over $\mathbb{Z}_4$, and Hensel's Lemma. The lemma is as follows: Let $f(x) \in \mathbb{Z}_4[x]$. Suppose $\mu(f(x)) = h_1(x)h_2(x) \cdots h_k(x)$, where $h_1(x), ...
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58 views

Polynomial restricts to zero on a hyperplane implies divisibility

$F$ is a finite field and $p \in F[x_1, x_2, \ldots ,x_n]$ is a polynomial If $p$ restricts to $0$ on a hyperplane $q = a_0 + a_1 x_1 + \cdots + a_n x_n = 0$ in $F^n$ then does it follow that $q$ ...
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100 views

Largest power of a prime dividing $q^m-1$

For positive integers $x$ and $d$ let $v_d(x)$ be the largest power of $d$ dividing $x$. Let $q>1$, $m$ a natural number, and $l$ a prime dividing $q-1$. Then I want to show that ...
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263 views

General advice on multiplying polynomials in a finite field?

Any tips on how to write the multiplication table in general for a finite field of polynomials (specific example: $F = (\mathbb{Z}/2\mathbb{Z})[x]/(x^2+x+1)$. I know that $F$ has four elements here, ...
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64 views

Is there a field with $n$ elements for all $n \in \mathbb{N}$? [duplicate]

I don't think this is true, but I'm not sure. I certainly know of finite fields with 2,4 and 8 elements, and of course $p^n$ elements where $p$ is prime, for all $n \in \mathbb{N}$.
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Field projection

I need a projection from $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{n-1}}$. I was thinking in a projection of vector spaces, but i want to know if there is a "canonical" projection or something like ...
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76 views

How can commuting with Frobenius imply the order of an element in the inertia group.

In this video, one asserts, in the beginning, that, for $\tau\in \mathbb V_0$ such that $\tau$ generates $V_0/V_1$ in the quotient group, and $\sigma\in \mathbb Z$ which is a Frobenius in $\mathbb ...
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135 views

Help with a bilinear form

Let $a,b\in\mathbb{F}_{2^{m}}$ (a field of characteristic $2$, m is odd) I need to prove that ...
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126 views

Trace function equation

Let $a,b\in\mathbb{F}_{2^{m}}$ (a field of characteristic $2$, m odd), with $a,b\neq 0$. I need to prove that $$\sum_{i=1}^{(m-1)/2}\operatorname{tr}(a^{2^{i}}b+b^{2^{i}}a)=0\qquad \text{ iff }\qquad ...
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491 views

Calculating Splitting Field Degree of Extension

Is there an easier way to calculate the degree of extension of a splitting field for a polynomial like $$x^7-3\quad\text{over}\quad\Bbb Z_5\,?$$My approach for several of these have been to find all ...
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116 views

The mathematics behind Sobol sequences

I am using Sobol sequence as random number generator in a computer program. Beyond just making the program work, I would like to learn the mathematics behind the Sobol sequence (and other ...
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3k views

Understanding Primitive Polynomials in GF(2)?

This is an entire field over my head right now, but my research into LFSRs has brought me here. It's my understanding that a primitive polynomial in $GF(2)$ of degree $n$ indicates which taps will ...
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1answer
135 views

Modules for twisted polynomial ring and Galois descent

Let $\mathbb{F}_q$ be a finite field with algebraic closure $\overline{\mathbb{F}}_q$ and consider the twisted polynomial ring $\overline{\mathbb{F}}_q\{ \tau \}$, where multiplication satisfies the ...
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Principal divisors

How can i calculate the principal divisor $(f)$ where $$f = \frac{(x^{3}-1)}{(x^{4}-1)}$$ with $f\in\mathbb{F}_2(x)$. I am recently reading about the subject, so i am looking for a simple solution ...
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452 views

Splitting field of $ x^2 + 1$ over $\mathbb{Z_3}$

I have the following exercise: Find splitting field for the polynomial $x^2 + 1$ over $\mathbb{Z_3}$. My solution: At first, we should try to solve the equation $x^2 + 1 = 0$, thus $x^2 = 2$ ...
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70 views

Extension field

Let $E$ an extension field of $k$ of grade $n$. I want to know if for $\alpha\in E$ the minimal polynomial of $\alpha$ has degree $r\leq n$. I think is true, but i could use some help
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Counting points in $\mathbb{F}_{p^n}$

Is there a closed formula for the number of elements of $\mathbb{F}_{p^n}$ which are not in any proper subfield of $\mathbb{F}_{p^n}$?
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Problem with roots of unity

Let $\zeta$ a root of $x^{p}-1$, with $p$ an odd prime, and $K$ a subgroup of the mutiplicative group $\mathbb{Z}_p^{*}$ of index $2$. I need to prove that $a=\displaystyle\sum_{k\in K}\zeta^{k}$ ...
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182 views

Cyclotomic polynomials, irreducibility [duplicate]

I need to decide if certain cyclotomic polynomials are irreducibles over the $\mathbb{F}_q$. For example, if $\Phi_{12}(x)$ is irreducible over $\mathbb{F}_9$. Anyone can help me? Ok, i think i ...
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Can extending a finite ground field make modules isomorphic?

$\def\Hom{\mathrm{Hom}}$Let $k$ be a field, $A$ a $k$-algebra and let $M$ and $N$ be $A$-modules, finite dimensional over $k$. Let $K$ be an extension of $k$, so $A \otimes K$ is a $K$-algebra and $M ...
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69 views

Commutative Algebra over a Finite Field

Let $A$ be a commutative algebra over the finite field $\mathbb F_q$, of order $q$, where $q=p^l$, for a prime $p$ and a positive inteher $l$. Assume that $A$ is finite-dimensional. So, $A$ is a ...
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496 views

Polynomial factorization over finite fields

How can i factorize the polynomial $x^{12}-1$ as product of irreducibles polynomials over $\mathbb{F}_4$? Anyone can help me?
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481 views

Irreducible cyclotomic polynomial

I want to know if there is a way to decide if a cyclotomic polynomial is irreducible over a field $\mathbb{F}_q$?
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592 views

Formula for number of solutions to $x^4+y^4=1$, from Ireland and Rosen #8.18.

There is a sequence of three exercise in Ireland and Rosen's Introduction to Modern Number Theory, Chapter 8, page 106. I can do the first two, but can't finish the third. I can include the proofs to ...
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179 views

Generators of fields, extending groups to fields, finite abelian groups

So I'm working through Koblitz'z "a course in number theory and cryptography" when I came across his proof that every finite field has a generator (ie, There is an element such that the multiplicative ...
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1k views

Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$

Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$ where $p$ is a prime. I'd like to start off by acknowledging that I know there are many posts relating to similar ...