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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clarification on Taylor's Formula

In Linear Algrebra form Hoffman and Kunze, the Taylor's Formula is stated as follows: Theorem 5. (Taylor's Formula) (page 129) Let $\mathbb{F}$ be a field of characteristic zero, $c\in \mathbb{F}$, ...
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Given an $n$, are there infinitely many finite fields $F$ such that none of the orders of $PSL_n(F)$ divide each other?

Given an integer $n$, is there an infinite set of finite fields $F_i$, $i\in \mathbb{N}$ such that for $i\neq j$ we have that $|PSL_n(F_i)|$ does not divide $|PSL_n(F_j)|$. The motivation is that ...
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Generating function identity from number of irreducible monic polynomials in $\mathbf{GF}(q)$.

I denote by $m_n(q)$ the number of irreducible monic polynomials with degree $n$ over the finite field of order $q$. So the number of monic polynomials with degree is just $q^n$. From this, how does ...
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131 views

How does one endow $\mathbb GF(p)^{n}$ with a field structure?

I've tried to endow $\mathbb{F}^{n}$, where $\mathbb{F}=\mathbb GF(p)$ with a field structure, but I was not able to do it. Could you please help me with this question?
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Easiest way to perform Euclid's division algorithm for polynomials

Let's say I have the two polynomials $f(x) = x^3 + x + 1$ and $g(x) = x^2 + x$ over $\operatorname{GF}(2)$ and want to perform a polynomial division in $\operatorname{GF}(2)$. What's the easiest and ...
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Is a field perfect iff the primitive element theorem holds for all extensions, and what about function fields

Let $L/K$ be a finite separable extension of fields. Then we have the primitive element theorem, i.e., there exists an $x$ in $L$ such that $L=K(x)$. In particular, the primitive element theorem ...
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456 views

Frobenius automorphism

Suppose $ q = p^r$. Let $F$ be the splitting field of $X^q - X$. Let $\phi : F \to F$ be the Frobenius automorphism $\phi(x) = x^p$. Then let $F' \subseteq F$ be the fixed field of $ < \phi^r ...
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106 views

Connecting the $+$ with the $\cdot$ operation in a field $(F,+,\cdot)$

I have to prove that the product of all invertible elements in a finite field,$F^*$, equals $-1$. Now I know that $F^*$ is cyclic, so taking the product over $1,l,l^2,\ldots,l^{|F^*|}-1$, where $l$ ...
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121 views

Question about odd and even part of polynomial

I am reading a paper, and have a next expresion Let $\sigma(X) \in F_{2^m}[X]$ with deg $(\sigma(X)) = \delta$, then writing $\sigma_1$ for the even part of $\sigma$ and $X\sigma'$ for the odd ...
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317 views

Proof existence of field extension of $\mathbb{F}_p$ containing the $r$-th primitive root of unity

I have to show the following: Let $p$ be a prime and $r \in \mathbb{N}$ with $\gcd(r,p)=1$. Prove the existence of a field extension $E$ of $\mathbb{F}_p$ which contains an $r$-th primitive root ...
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371 views

Product in GF(16)

i need some help with a product in GF(16), where it is seen as an extension of GF(4)={0, 1, x, x+1} (where $x^2 = x + 1$ ) with the irreducible polynom $f(y) = y^2 + y + x$ So the elements in the ...
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1k views

Frobenius Automorphism as a linear map

Let $\phi(x) = x^p$ be the Frobenius automorphism on $\mathbb F_{p^n}$. We can view $\mathbb F_{p^n}$ as an $n$-dimensional vector space over $\mathbb F_p$. In this case, $\phi$ is a linear ...
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558 views

Necessary and Sufficient Condition for a sub-field

Is there any necessary and sufficient condition to determine whether a subset $H$ of a given field $K$ is a subfield? In some paper I have found something like that: $H$ is a field if for all $a, ...
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423 views

Squarefree polynomials over finite fields

I'm trying to figure out how many squarefree polynomials there are of a fixed degree over $\mathbb{F}_2$ specifically (and in general, over any finite field). Looking at some low-degree examples seems ...
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218 views

Is correcting 2 consecutive error's in 9 messages from $ GF(2^6) $ by turning tham into 3 messages and solving Reed-Solomon code $ (3, 18) $ possible?

2 consecutive messages have errors. We have 9 messages from $ GF(2^6, x^6+x+1) $. Messages were encoded with $ (x+1)(x+a)(x+a^2)\sum_{i=1}^6X_ix^{i-1}=\sum_{l=1}^9Y_lx^{l-1} $ , where $ ...
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128 views

How to decode encoded, and corrupted in transmission message in Galois Field $2^5$ with one error?

We are given $GF(2^5, x^5+x^2+1 )$. We had some $ X_1, ..., X_5 $ message items from our $ GF(32) $ which we do not know and need to find. Thay were encoded via service blocks $ Y_1...Y_7 $ with next ...
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236 views

Is there any field of characteristic 4 or any other composite number? [duplicate]

Possible Duplicate: Characteristic of a field is $0$ or prime Is there any field of characteristic 4? Or any other composite number?
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Constructing an explicit isomorphism between finite extensions of finite fields

Suppose $K$ is a finite field, $K = \mathbb F_{p^s}$. If we take an irreducible polynomial $f$ of degree $d$ over $K$, then the splitting field $L$ of $f$ is $K(\alpha)$ where $f$ is the minimal ...
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3answers
119 views

finite fields factorization

Let $\mathbb{F}_2$ be the finite field with two elements. Let $f(x) = x^6+x^4+x+1$ be in $\mathbb{F}_2[x]$. If $f(x)$ is irreducible, give a reason. If it is not irreducible, determine a factorization ...
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Splitting fields of polynomials over finite fields

I can't follow a statement in my notes: "Let $K$ be a finite field, with $f \in K[X]$ an irreducible polynomial of degree $d$. Then any finite extension $L/K$ is normal, and so if $L$ contains one ...
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573 views

Isomorphism of sets

What is an isomorphism of sets? I know in general an isomorphism is a structure-preserving bijective map between two algebraic structures. But what algebraic structure does a set have? Does a ...
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Addition and multiplication in a Galois Field

I am attempting to generate QR codes on an extremely limited embedded platform. Everything in the specification seems fairly straightforward except for generating the error correction codewords. I ...
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Count points and lines in $\mathbb{A}^2(\mathbb{F}_p)$

Let $p$ be a prime, then $\mathbb{F}_p$ is a finite field. $\mathbb{A}^2(\mathbb{F}_p)$ is an affine plane. Number of points in $\mathbb{A}^2(\mathbb{F}_p)$ is $p^2$. I look at a line equation ...
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68 views

Irreducibility in $\mathbb{F}_3[x]$

How do we know if something is reducible/irreducible in $\mathbb{F}_3[x]$ in terms of polynomials?
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93 views

What does the notation $\mathbb{F}_3[x]$ mean?

What does $\mathbb{F}_3[x]$ mean in terms of polynomials?
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Why is it that in $\mathbb{F}_q \setminus 0$, there are exactly as many squares as non-squares?

Why is it that in $\mathbb F_q\setminus\{0\}$, when $q\neq 2^k$, there are exactly as many squares as non-squares? My combinatorics textbook states this without proof :/
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370 views

Infinite Field of Characteristic 5

I recently took an exam in which the professor asked to give an example of an infinite field of characteristic 5. I had studied this problem, and found examples such as this. My answer that I ...
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Are quartic minimal polynomials over $\mathbb{Q}$ always reducible over $\mathbb{F}_p$?

This situation arose while studying biquadratic extensions. Let $\mathbb{Q}(\alpha)$ is some biquadratic extension, with $m(x)$ the minimal polynomial of $\alpha$. Suppose that ...
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A ring with a subring that is a field

I'm looking for an example of a ring $R$ such that $R$ has no multiplicative identity, but R has a subring $A$ which is a field
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Question about LFSR

I am reading a paper and say this "The idea is to load $f(X)$ into LFSR to multiply by $X$ mod $g(X)$(primitive polynomial deg $g=n$). We next compute a polynomial h(X) whose coefficients are given ...
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Prove that $F[x]/(x^2+1)$ has 121 elements

$F$ is the field of integers mod 11. How can I show that $F[x]/(x^2+1)$ has 121 elements? Though,it is intuitively clear because there are 121 one degree polynomials, but how do I prove it ...
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Is there an explicit description of the fields laws on this field?

I'm working on a fairly simple problem about a field, but I want to know if the operations can be explicitly described. Suppose $c$ is not a quadratic residue modulo $p$, and consider the quotient ...
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On the order of elements of $GL(2,q)$?

There's a particular property of the elements of $GL(2,q)$, the general linear group of $n\times n$ matrices over a finite field of order $q$, that I don't understand. First, I know that the order of ...
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Example of a reduced ring over a finite field satisfying some other conditions

What is an example for: An extension of rings $k \subset R$ where $k$ is a finite field, $R$ is a finite dimensional vector space over $k$, $R$ is reduced, and $R \neq k[r]$ for all $r \in R$. So, ...
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Solve inequations over a finite field

The following is a special case of my earlier question which is still not solved. Suppose both $\mathbf{a_i}$ and $\mathbf{v}$ are $1\times N$ vectors over a finite field $\mathbb{F_q}$, where $i\in ...
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characteristic of a finite field

knowing that the characteristic of an integral domain is $0$ or a prime number, i want to prove that the characteristic of a finite field $F_n$ is a prime number, that is $\operatorname{char}(F_n)\not ...
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reducible polynomial modulo every prime

how to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$. For example i know that $\mod 2 $, $x^4+1=(x+1)^4$ . Also $\mod 3$,we have that $0,1,2$ are not ...
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Example of matrix $M\in GL_3(\mathbb{Z}/7\mathbb{Z})$ such that $\langle M; +, \cdot \rangle \simeq GF(7^3)$ and the multiplicative order of $M$ is 3

I would want to make an example of a matrix $M \in GL_3(\mathbb{Z}_7)$ such that $\langle M; +, \cdot \rangle \simeq GF(7^3)$ and the multiplicative order of $M$ is $3$. Any hints how to do that ...
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Reed Solomon Polynomial Generator

I am developing a sample program to generate a 2D Barcode. And i am using reed solomon error correction code. By Going through this article i am developing the program. But i couldn't understand how ...
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779 views

A Hunt for a Mathematical Machine That Gives Points

The central question is : Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ? Explanation: ...
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What is Galois Field

When I am reading some algorithm related to error correction. To generate some polynomial it uses finite-field which is Galois field. I am not from mathematical background. Can anybody explain me in ...
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Order of finite fields is $p^n$

Let $F$ be a finite field. .How do I prove that the order of $F$ is always of order $p^n$ where $p$ is prime?
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A resemblance between 2 binomial identities - why?

Let $F$ be any field (or even ring). The following formal power series identity (i.e., equality in $F[[x]]$) holds for any $j \ge 0$: $$(1-x)^{-j} = \sum_{i \ge 0} \binom{i +j -1}{i} x^i $$ The ...
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Low degree approximation of the polynomial extension of the logical-or function

Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$). Consider ...
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Extending the logical-or function to a low degree polynomial over a finite field

Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$). Is there ...
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Probability of a random $n \times n$ matrix over $\mathbb F_2$ being nonsingular

Given a random square matrix of size $n\times n$ in the field $\mathbb F_2$, what is the probability that its determinant is $1$? (This is also the probability that the matrix is non-singular, since ...
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Low degree extension

Let $v$ be a binary vector of dimension $n$. Assume that $n$ is a perfect square, then $v$ can be thought of as a function $f:[\sqrt n]\times[\sqrt n]\to \{0,1\}$, where $[\sqrt n]=\{1,\ldots,\sqrt ...
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Irreducible polynomial of $\mathrm{GF}(2^{16})$

I'm implementing some code for the Galois field $\mathrm{GF}(2^{16})$. Which irreducible polynomial do you recommend that I use?
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Explicit examples of infinitely many irreducible polynomials in k[x]

My question is the following. Is it possible to give examples of infinitely many irreducible polynomials in a polynomial ring $k[x]$ with $k$ a field? I'm interested in this because I'm ...
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Nilpotent matrices over finite fields

What sort of criterion is there for determining whether a matrix is nilpotent? Specifically, I'm interested in the nilpotent matrices over finite fields. I realize that any such matrix will have to ...