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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How to get the irreducible polynomial?

I am struggling with the finite field. It is easy to work out all the irreducible polynomials. polynomials when the order is low say 3,4 over $\mathbb F_2$. But is there an easy way to write out all ...
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Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
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Cyclic Codes over $GF(q)$

Does the set of cyclic codewords / codeword polynomials themselves form a field ? I think they donot because the modulo operation is with respect to $x^n-1$ which is not a prime polynomial. Also the ...
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Who invented the standard construction of finite fields (and field extensions)?

The title says it all, but just to be clear I mean the construction of taking $k[x]$ modulo an irreducible polynomial of suitable degree. Was it an open problem for any considerable amount of time, ...
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217 views

Properties of a sum over the root-of-unity expression of polynomials over a finite field

Consider a bivariate polynomial over the finite field $\mathbb{Z}_n$ of the form: $$f(x,y) = c\cdot xy + g(y)$$ where $c$ is some non-zero constant and $g$ is some univariate polynomial. Let ...
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Prime polynomials over GF(q)

Could anyone please throw some light on my questions? What is the splitting field of an $m$th-degree prime polynomial over $GF(q)$? Is it always $GF(q^m)$? How many $m$th-degree prime polynomials ...
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Proving the roots of a polynomial are a subfield

Define $f(X)=X^{p^r}-X \in \mathbb{F}_p[X]$ where $r\geq 1,\ p$ prime. I'm to show its roots are a subfield of its splitting field $E$ (I have already shown it has $p^r$ distinct roots in $E$). What ...
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507 views

Eigenvalue problems for matrices over finite fields

Suppose I have a symmetric matrix A with entries in a finite field. In particular, I have the case in mind where $A \in \{0,1\}^{n \times n}$ and want to treat the entries as elements of $GF(2)$. How ...
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51 views

Show that all irreducible polynomials divide $t^{p^d-1} -1$

I have shown that if $f$ is irreducible with $deg(f) = d$ then $\mathbb{F}_p [t] / \langle f\rangle$ is a field with $p^d$ elements. I've also shown that $\langle f\rangle$ is precisely those ...
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Question about why $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$.

I'm trying to follow a solution I'm reading. The idea is to prove $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$ when $n=1$ or $n=p=2$. The solution is as follows: If $x^{p^n}-x+1$ is ...
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125 views

Probability and Rank of Symmetric Matrix

Let be a $n \times n$ Symmetric Matrix over Finite Field with $q$ elements, where the green color represents 0's and the black color non-zero entries. How I will be able to demonstrate that ...
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75 views

Factoring $x^{16}-x$ over $\mathbb{F}_8$

A homework question asks me to factor $x^{16}-x$ over the finite fields $\mathbb{F}_4$ and $\mathbb{F}_8$. I got the result for $\mathbb{F}_4$ using the factoring over $\mathbb{F}_2$ and then a ...
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100 views

E: $y^2+y=x^3$ an elliptic curve over $F_{2}$. How to prove the number of $E(F_{2^n})$ = $2^n+1$ if n is odd, …

Let E be the elliptic curve $y^2 + y = x^3$ over $F_2$. Prove $ $#E($F_{2^n})$$ = \left\{ \begin{array}{ll} 2^n+1 & \quad n=odd \\ 2^n+1-2(-2)^{n/2} & \quad ...
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How hard is finding values such that

We can work with powers of some naturals $(x_k)^{m_k}$. Here we have $n$ naturals, and $m_k$ is an integer in the range $-r$ to $r$. My question is, how small can $p$ be so that ...
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79 views

Cyclic Rotation over Finite Fields

Let $\mathbb{F}$ be a finite field with $q$ elements and $\mathbb{E}$ an extension field of degree $n$ of $\mathbb{F}$. How I will be able to demonstrate that ...
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112 views

Calculating characteristic polynomials of matrices in GF(2)

How do you calculate a characteristic polynomial of a matrix in GF(2)? I understand the concept of characteristic polynomials in matrices using "regular" math with real numbers, but I'm a bit confused ...
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100 views

Rank of Quadratic Form

Let $n,m, s \in \mathbb{Z}$ be integers satisying $n=s^2$ and $m=2n$. Let $\newcommand{\bigmatrix}[1]{ \begin{pmatrix} #1_1 & #1_2 & \cdots & #1_s \\ #1_{s+1} & #1_{s+2} & \cdots ...
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39 views

Difference between Galois and other automorphisms

With respect to the definition of Galois field, for $E$ an extension of $F$ ($E$ and $F$ are finite fields) $\mathrm{Gal}\,(E/F)$ is the set of automorphisms of $E$ which fix $F$ pointwise. So I think ...
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85 views

Theory of Fields $\omega$-Inconsistent?

A theory is $\omega$-inconsistent if there is a predicate $P(n)$ that is true for every standard natural number yet not true for all numbers. Consider the theory of finite fields and let $P(x) = (Sx ...
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59 views

generating set for finite fields

let us consider GF(2^n) as a vector space over GF(2), Is it possible to find a generating set for GF(2^n)? How can ew find it? I want to define a linear transformation of GF(2^n) to itsefl.
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136 views

elementary abelian groups and finite fields

in group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order $p$, where $p$ is a prime; it is a particular kind of $p$-group. now suppose that we ...
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159 views

System of linear equations over Finite Field with restriction on variables

Given two vectors $x$ and $y$, where each element $x_i, y_i$ is from a finite field. I have the restriction that for each of these variables only about half of the elements of this finite field are ...
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Binomial Coefficient Finite Field

Let $\mathbb{F}$ be a finite field with $q$ elements and $\mathbb{E}$ an extension field of degree $n$ of $\mathbb{F}$. Let $S(x) \in \mathbb{E}[x]$. How I will be able to reduce a expression: ...
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Discover parameters of a Reed-Solomon code from its output

chirp.io is a site/app for sharing e.g a photo identified by a short FSK audio chirp. The chirp is 10 symbols of data, then 8 symbols of error correction. Thes symbols are 32-valued (5 bits/symbol) ...
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112 views

Irreducibility of $x^{2^n}+x+1$ over $\mathbb{Z}_2$ [duplicate]

I'm trying to solve this problem from Hungerford V.5.9. I have to show $x^{2^n}+x+1$ is irreducible over $\mathbb{Z}_2$ if n>2. I would appreciate some hint cause I don't know how to start with ...
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automorphism of fields

let we consider $GF(p^n)$ as a vector space over $GF(p)$, $p$ is prime. Also we want to have an invertible linear map on $GF(p^n)$, (automorphism of $GF(p^n)$). on the other hand we know that A field ...
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Is vectorspace trivial under these conditions?

Let $R$ be a ring. Looking for a left $R$-module free over abelian group $A$, I arrived at $\left|R\right|\otimes A$ with $r.\left(s\otimes a\right)=rs\otimes a$ where $\left|R\right|$ denotes the ...
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Field of order 8, $a^2+ab+b^2=0$ implies $a=0$ and $b=0$.

I was able to come up with a proof for this problem however, it seems like my argument can work for any field of even order and not just odd powers of 2 so I'm convinced there is something wrong here. ...
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How do we distinguish between characteristic 0 and characteristic p for very large p?

This is a somewhat soft question, apologies if it turns out to be trivial/nonsensical. Background: I was half-asleep one morning, not quite through my first cup of coffee, and thought about the ...
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How can I find the distribution of a recursive relation with two parameters?

Suppose we have a recursive relation, e.g. $G(n,m) = G(n-1,m) + G(n, m-1)$, with some initial points where $n,m \in \mathbb{Z}^{+}$ and $F$ is a finite-field, e.g. $\mathbb{Z}_p$ for a prime $p$. Also ...
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How $x^9-1$ splits over $\mathbb{F}_p$

Let $p\equiv -1 \pmod 9$ be a prime. Determine the factorization of $f(x)=x^9-1$ over $\mathbb{F}_p$. My proof: First of all we observe that $f(x)=(x-1)(x^2+x+1)(x^6+x^3+1)$. The degree of the ...
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An equation over $\mathbb{F}_{2^m}$

Let $m$ and $k<2^m$ be nonnegative integers and let $\alpha \in \mathbb{F}_{2^m} - \mathbb{F}_2$ such that $$ \left\{ \begin{array}{l} \alpha = \alpha^k \\ \alpha + 1 = (\alpha + 1)^k \end{array} ...
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Subgroups of Galois groups of finite fields

According to the notion of Galois group, for $E=GF(2^n)$ as an extension of the field $F=GF(2)$, the Galois group $Gal(E/F)$ is a cyclic group of order $n$. Now my question is: for finding the ...
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Finite fields, Linear Algebra

I just started my first upper level undergrad course, and as we were being taught vector spaces over fields we quickly went over fields. What confused me was when the book (Advanced Linear Algebra, ...
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Univariate and Matrix Representation of Affine Transformation

Let $\mathbb{F}$ be a finite field with $q$ elements and $\mathbb{E}$ an extension field of degree $n$ of $\mathbb{F}$. Let $S:\mathbb{F}^n\rightarrow \mathbb{F}^n$ be a affine transformation and ...
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Why is the answer set limited here?

This question is based on pp $67$ - $68$ of Ash and Gross's "Elliptic Tales". Here the authors discuss points on a curve in the projective plane. We have an equation $f(x,y) = x^2+y^2$ We can ...
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126 views

tensor product of a vector space and finite field

I want to know how we can interpret and define the tensor product of V as a vector space with a F as a finite field?
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kernel of maps associated to the root of an irreducible polynomial

Let $m(\mu)$ be an irreducible polynomial of degree $d$ over $\mathbb{F}_2$, $F_{2^d} = \mathbb{F}_2[x]/(m(\mu))$ by a field extension given by that polynomial and let $d: \mathbb{F}_2[x] \to ...
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Factor polynomials into irreducibles over GF(q)

The polynomials $x^4+x^3+1$ and $x^4+x^3+x^2+x+1$ are irreducibles over GF(2). (a) Factor both polynomials into irreducibles over GF(4). (b) Factor both polynomials into irreducibles over GF(8). I ...
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What is the connection between Weil's character bound and Riemann Hypothesis over finite fields

Weil's character bound states that: Let $\mathbb{F}_{q}$ be a finite field of size $q$. Let $\chi$ be a multiplicative character of order $m$. Let $f(x)$ be a polynomial of degree $d$ such that $f(x) ...
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probability of choosing $k$ linear independent elements from a finite field

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. What is the probability of choosing $k$ linear independet elements from $\mathbb{F}_q^n$? I guess, it depends on how we choose from ...
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Prove two bases are dual in a finite field.

Let K be a finite field, $F=K(\alpha)$ a finite simple extension of degree $n$, and $ f \in K[x]$ the minimal polynomial of $\alpha$ over $K$. Let $\frac{f\left( x \right)}{x-\alpha }={{\beta ...
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Prove a polynomial in Fq is a permutation polynomial of Fqn with a necessary and sufficient condition

P.S This is the best Math Expression I can edit. I am real shameful, where can I find the introduction of typing in this webset? thank you Exercise7.13 Let\[f\left( x \right) = \sum\limits_{i = ...
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Irreducible quadratic in $\mathbb{Z}_p$

I want to show that for every prime $p$, there exists an irreducible quadratic in $\mathbb{Z}_p[x]$. So I'm looking for some $x^2+ax+b$ that's irreducible. But what $a,b$ choose we choose?
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Quadratic Polynomials over $\mathbb F_p$

I'd like to know a reason why (irreducible) quadratic polynomials over $\mathbb F_p$ do not reach all numbers in $\mathbb F_p$. Example: $f(x)=x^2+3x+1$ in $\mathbb F_7$ is irreducible, i.e has no ...
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Show that a given ring is a field with four elements

Let $R = ( \mathbb{Z} / 2 \mathbb{Z} ) [t]$ be the ring of polynomials with coefficients $\mathbb{Z} / 2 \mathbb{Z}$, $f = f(t) = t^2 + t +1$, and $g = t^2 +1$. Show that: (1) $R/(f)$ is a field with ...
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245 views

software to factor polynomials over finite field extensions

I need a software/computer algebra package which allows irreducible factorization of a polynomial over finite field extensions (specifically $GF(2^{n}$)). After searching online I could only find ...
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My proof: Frobenius Map generates $\mbox{Gal}(\mathbb F_{p^n}/\mathbb F_p)$

I would like to ask, whether anyone can confirm or correct the following version of the proof that the Frobenius map generates the Galois Group of a finite field. Proof First note that ...
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Find the number of irreducible factors of $x^{63} - 1$

I have to find the number of irreducible factors of $x^{63} - 1$ over $\mathbb{F}_2$ using the $2$-cyclotomic cosets modulo $63$. Is there a way to see how many the cyclotomic cosets are and what is ...
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Low Index Subgroups in GAP

I want to find an algorithm for constructing a coset table for all subgroups of index 14 of triangle group (2,3,7) by using Low index subgroups. for this I am making a program like this. ...