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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Finding quadratic residues in a finite field by using a primitive element

Let $1+2x$ be a primitive element of the field $\mathbb F_9$ obtained via the irreducible polynomial $$x^2 + 1$$ over the base field $\mathbb F_3$. i) Make a list of the elements of $\mathbb F_9$ ...
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On any finite field, adding the identity element a finite amount of times will result to the neutral element

Show that if you add the identity element ($1$) a finite amount of times will result to the neutral element ($0$). I started with saying that in every field there's an element $a \in F$ so that $a + ...
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347 views

Lagrange interpolation of Galois field functions

I'm trying to understand how to apply Lagrangian interpolation for a function $F : GF(2^k) \to GF(2^k)$, but am having some difficulty. Suppose I have a function $F : GF(2^2) \to GF(2^2)$ given by the ...
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Order of matrices in $SL_2({\mathbb{F}_q})$

Could you tell me how to prove that in $SL_2({\mathbb{F}_q})$ the only element of even order is $-I$ ($ \ I$ - identity matrix)? I would really appreciate a thorough explanation, because I cannot ...
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Finite fields and primitive elements

Let $\mathbb F_9$ be a finite field of size $9$ obtained via the irreducible polynomial $x^2 + 1$ over the base field $\mathbb F_3$. How can you find a primitive element? Make a list of the ...
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Question about terminology, finite fields

My English is not very good, and that's why I would really appreciate it if you could explain to me what the phrase : these elements are under the same domain under $F$ and $\alpha$ means in this ...
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48 views

Number of solutions of $P(x_1, …, x_n) = 0$ in $\mathbb{F}_q^n$

I have this exercise : $q = p^r$ (it is not clearly precised in the exercise that $r = n$). and $\mathbb{F}_q$ is the finite field of cardinal $q$. Let's $P \in \mathbb{F}_q[X_1, ..., X_n]$ of degree ...
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Counting roots of a multivariate polynomial over a finite field

How many roots can there be of a polynomial $f \in K[x_1, x_2, \dots , x_n]$ where $K$ is a finite field and the maximum exponent of $x_i$ in any term is $m$ for all $i$, assuming not all coefficients ...
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79 views

Rational and irrational fractions over finite fileds [duplicate]

I've been told that over the field $\mathbb{F}_7$ the square root of $2$ is actually $3$. How come? Why does it happen?
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619 views

Roots of polynomials over finite fields

I've been trying to find the decomposition of $x^2-2$ to irreducible polynomials over $\mathbf{F}_5$ and $\mathbf{F}_7$. I know that for some $a$ in $\mathbf{F}_5$ (for example), $x-a$ divides $x^2-2$ ...
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54 views

Polynomials decomposition into irreduceables

I've been trying to find the composition to irreduceables of the following polynomials with no much success: X^2 +1 over the field F7 and X^2-2 over the field F5 Is there any method/algorithms I ...
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73 views

Maps compatible with the Frobenius

Let $F$ be a field. Fix a separable closure $F^{sep}$. Consider the monoid whose elements are maps of sets $F^{sep} \to F^{sep}$ which are equivariant with respect to the Galois action. These maps ...
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549 views

Minimal polynomial of a finite purely inseparable field extension

Given $F$, a field with characteristic $p > 0$, and a finite purely inseparable field extension $E$. Then prove that the minimal polynomial $f(x)$ of any $\alpha \in E\backslash F$ will be of the ...
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Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$

I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$. Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines. There are $q+1$ ...
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167 views

The linearity of the extended code

If we add a last digit to the code $C$ of length $n$, we obtain a new code called extended code. My question is: If the code $C$ is linear, can we prove that the extended code $C'$ is linear too? ...
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Are there any other constructions of a finite field with characteristic $p$ except $\Bbb Z_p$?

I mean, $\Bbb Z_p$ is an instance of $\Bbb F_p$, I wonder if there are other ways to construct a field with characteristic $p$? Thanks a lot!
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Show that A is a field which has $9$ elements

For $A=\left\{\left( {\begin{array}{*{20}{c}} a&b\\ { - b}&a \end{array}} \right) | a,b\in\mathbb{Z/3Z}\right\}$ . Show that A is a field which has $9$ elements . $(A^*, .)$ is a cyclic group ...
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Lin Alg- Dual Spaces

Let $(V^*)^*=V^{**}$. Define $S:V\to V^{**}$ by $s(v)(\alpha)=\alpha(v)$ for all $v\in V$ and $\alpha\in V^*$. I need to show that $s(v)\in V^{**}$. And show the S is a linear transformation. ($V$ is ...
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202 views

Discrete logarithm - strange polynomials

If $p$ is a prime number and $\omega$ is a fixed primitve root for $\mathbb{Z}/p\mathbb{Z}$, then we can define the discrete logarithm of $x \in (\mathbb{Z}/p\mathbb{Z})^{\times}$ as the unique number ...
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Polynomial $x^3- xy^3$ and the like over finite fields.

Let $f_{a,n}(x_1,x_2)$ be a polynomial in $\mathbb{F}_p[x_1,x_2]$, where $\mathbb{F}_p$ is a finite field or oder $p$ (perhaps, we may first assume that $p$ is prime) depending on $a\in\mathbb{F}_p$ ...
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111 views

extended linear codes over the field $\mathbb F_q$

Suppose we extend the $[n,k]$ linear code $C$ over the field $\Bbb F_q$ to the code $C'$, where $$ C' = \{(x_1,\ldots ,x_n,x_{n+1})\in \Bbb F_q^{n+1} : (x_1,\ldots,x_n) \in C \text{ and } ...
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Why is $X^4 + \overline{2}$ irreducible in $\mathbb{F}_{125}[X]$?

I want to prove that $f = X^4 + \overline{2}$ is irreducible in $\mathbb{F}_{125}[X]$. I know that $\mathbb{F}_{125}$ is the splitting field of $X^{125} - X$ over $\mathbb{F}_5$, and that this is a ...
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120 views

Finding number of solutions to an equation in $\mathbb F_p$

$p=3 \pmod 4$ is a prime. Let $b\in \mathbb F_p^*$. Show that the equation $v^2=u^4-4b$ has $p-1$ solutions $(u,v)$ with $u, v \in \mathbb F_p$. If we write the given equation as $v+u^2=x$ and ...
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79 views

Convert polynomials and fractions in a finit field?

I am trying to understand how finite field works, and I am stuck on converting high power polynomials into a power of the field, also converting fractions into integers. $8^{-1}\cdot44$ in $\Bbb ...
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114 views

How to list all irreducible polynomials in a field?

I am currently trying to refresh my memory on some basic primary polynomials, apologies if my terminologies aren't correct: For example, I have a field $\Bbb F_{2^3}$ and generates a list of ...
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639 views

Rank of matrices multiplication

Matrices $m_1$ and $m_2$ are over a finite field($GF(2^{8})$ for example). $m_1$ is a $m\times n$ matrix($n > m$) with $rank(m_1) = m$, and $m_2$ is a $n\times c$ ($c > n > m$) matrix ...
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612 views

A question about irreducible polynomials, roots, and polynomial bases

I have a couple questions about the mathematical jargon that's thrown around in the literature all too often. To my knowledge, an irreducible polynomial is one that cannot be factored, meaning that it ...
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55 views

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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88 views

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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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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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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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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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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294 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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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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961 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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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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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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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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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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$E/\mathbb F_q$ 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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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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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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201 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 ...