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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Finite fields and characteristics [duplicate]

I am just a little confused about finite fields and fields with characteristic $p$ (prime number). I want to know if the following proposition is true: A field $F$ is finite if and only if it is of ...
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Finding primes such that a given polynomial is irreducible modulo $p$

Let $f \in \mathbb{Z}[x]$ be irreducible, and let $\bar{f} \in \mathbb{F}_{p}[x]$ be the image of $f$ in the polynomial ring over the finite field with $p$ elements. Is there a general procedure, ...
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False proof that $F_{3^2}$ contains $F_{3^4}$

Let's say we adjoin a second degree algebraic number to $F_3=\mathbb Z/3\mathbb Z$, it doesn't matter which. Then we get a field of $9$ elements, $F_{3^2}$. On the other hand, if we adjoin a fourth ...
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Algebra Splitting fields

Take the irreducible polynomial $x^3 + x^2 + 1$ over $F_2$ (field of order $2$). Find the splitting field and its roots in that field. Where I am: I understand what splitting fields are, and I ...
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Find the number of disjoint cycles of the map $\phi_q: \Bbb{F}_{q^p}\to \Bbb{F}_{q^p}$

Let $\Bbb{F}_q$ be a finite field. I need to find the number of disjoint cycles of the map \begin{align*} \phi_q: \Bbb{F}_{q^p}&\to \Bbb{F}_{q^p}\\ \alpha &\mapsto a^q, \end{align*} where ...
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Can we give efficiently the solution of a system of bilinear equations over a finite field?

Consider a finite field $F$ and suppose we have a system of equations $$h_1(\alpha,\beta)=0,h_2(\alpha,\beta)=0,...,h_t(\alpha,\beta)=0$$ where $\alpha=(\alpha_1,...,\alpha_s)$ and ...
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Is there any isomorphism between the quotient ring $F{_p}$ /$\left\langle {{x^n} - a}\right\rangle$ and the group algebra $F{_p}G$

We know that the quotient ring $F{_p}$ /$\left\langle {{x^n} - 1}\right\rangle$ is isomorphic to the group algebra $F{_p}C{_n},$ where $F{_p}$ is a finite field of characteristic $p$ and $C{_n}$ is a ...
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Show the following system is not possible

Assume throughout that the base field is the prime field $\mathbb{F}_2$. I have two $n \times n$ matrices: $I_n$, the $n \times n$ identity matrix, and $C_n$ the matrix obtained from $I_n$ by shifting ...
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60 views

Existence of Field with $p^n$ Elements.

If $p^n$ is a prime power, how can we show that there exists a field $F\supseteq\mathbb{F}_p$ such that $$x^{p^n}-x=(x-\theta_1)(x-\theta_2)\cdots (x-\theta_{p^n})$$ for some $\theta_i\in F$? Using ...
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How to prove that the additive group of a finite field of order $p^n$ is isomorphic to $Z_p^n$?

Let $\mathbb{F}$ be a finite field of cardinality $p^n$ where $p$ is prime. How to prove that the additive group of $\mathbb{F}$ is isomorphic to $Z_p^n$?
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How is integer polynomial factorization reduced to factorization over a finite field?

I've read on Wikipedia that the problem of factoring polynomials over $\mathbb Z$ can be reduced to factoring polynomials over some finite field, but I can't find any information on how this is done. ...
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69 views

If $E,F$ are finite fields and $F\subseteq E,$ why is $E$ a finite-dimensional vector space over $F$?

I understand that if $E$ and $F$ are each finite and $E$ is a vector space over $F$, then $E$ must be a finite-dimensional vector space over $F$. However, my question is: why does $F\subseteq E$ imply ...
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Prove the polynomial is irreducible depending on the order of a field

I have to show that $f(x) = x^4+x^3+x^2+x+1$ is a irreducible polynomial in $F_p$ with $p \equiv 2 \pmod{5}$ or $p \equiv 3 \pmod{5}$. $f(x) \mid (x^5-1)$. This should be used for order of possible ...
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18 views

Unusual syntax for finite field

I found in an exercise the following syntax: Polynomial f(x) =..... is a polynomial in Fp with p = 2 mod 5 or p =3 mod 5. I have to show that it is irreducible. I don't know this Syntax p= 2 mode 5 ...
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39 views

Find the degree of the polynomials in the following groups

Let $f(x) = x^4 + 6x^3 + 15x^2 + 10x + 1$ and $g(x) = 2x^2 + 15x + 1$. Consider $f$ and $g$ as polynomials with coefficients in (a) $\mathbb Q$, (b) $\mathbb F_2$, (c) $\mathbb F_3$, and (d) $\mathbb ...
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Trace in finite Fields

(1) In $\mathbb{F}_{2^n}$ with odd n it should be shown that half of the Trace values are 0 and the other part is 1 with help of the additivity of the Trace. (2) Now n shall be even. Now the ...
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16 views

Separability of $X^{q^d}-X\in\mathbb F_q[X]$ for $d\in\mathbb N$

I know that I can verify that $f:=X^{q^d}-X\in\mathbb F_q$ and $f'$ (the formal derivative) are coprime in order to establish $f$'s separability. Is there an easier way, in particular one that does ...
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55 views

Find irreducible polynomial over $\mathbb F_9$

I am looking for a polynomial of degree $3$ in $\mathbb{F}_{9}$. How do I find one ? And if I have one how do I show that it is irreducible ? I would start with an irreducible polynomial in ...
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40 views

Elementary algebraic question about finite-fields.

I am struggling to understand what the following question is asking me to prove. In particular, I am not understanding whether the "p" in $\mathbb{Z}/p\mathbb{Z}$ is a prime or something else ...
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29 views

Polynomial to polynomial function on (in)finite field [closed]

Let K be a field. Prove that a transformation K[x]->(polynomial functions K->K) is injective if and only if K is an infinite field. How do I approach it? It's probably a very simple problem cause ...
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23 views

Change of factorization in extension field

I have to factorize the polynomial ($x^8-x$) in $\mathbb{F}_{2}$. I found the following factorization: ($x^8-x$) = x*($x+1$) * ($x^3$+x+1)* ($x^3$+$x^2$+1). But now I change to the $\mathbb{F}_{4}$. ...
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Factoring Polynomials in Fields

I always have problems to factorize polynomials that have no linear factors any more. For example ($x^5-1$) in $\mathbb{F}_{19}$. It's easy to find the root 1 and to split it. ($x^5-1$) = ($x-1$) * ...
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Confused about Finite fields and polynomials

I'm asked to give a polynomial that has a root over a finite field but not a root over R. My understanding is that the finite field is contained in R (more restrictive) so how can there be a root in ...
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28 views

Clarification on rings of polynomials / Galois fields

I need some clarification about what the following thing is called (it's commonly used to describe linear-feedback shift registers), and what a good computer program identifier would be that describes ...
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53 views

Factorization of $x^5-1$ over $F_{11}$ and $F_{19}$

I got the following question. The polynom $x^5-1$ should be factorized over $F_{11}$. I have this as a first solution: $x^5-1=(x-1)(x+1)(x^2+1)(x-1)$ could this be possible? I don't know which ...
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35 views

Does this equation in $\mathbb{Z}_p$ always have a solution?

Is there always a solution to the equation in the field $\mathbb{Z}_p$ ($p$ being a prime number) $$ a^2 + b^2 \equiv c \pmod p $$ for a given $c \in \mathbb{Z}_p$? The solution need not be unique, ...
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69 views

Points on an elliptic curve over $\mathbb F_p$

Let $E$ be an elliptic curve over $\mathbb F_p$ (the finite field of $p$ elements) defined by $y^2=x(x-n)(x-m)$ where $p\nmid nm(n-m)$. Let $N$ be the number of $\mathbb F_p$-valued points of $E$. ...
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Bijections of a finite field that preserve the kernel of the trace

Let $q=p^n$, for some prime $p$ and positive integer $n$. Let $m$ also be a positive integer, and denote $Tr$ to be the trace of $GF(q^m)$ over $GF(q)$. I have the suspicion that all the functions ...
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How to find all possible polynomials over a given finite field?

How would I find the possible polynomials over GF(p)? I'm trying to figure out which polynomials of a specific given finite field have no roots.
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59 views

Matrices such that their sum equals their product

Let $N > 1$ and consider square matrices of size $N$. Let $J$ be the matrix full of $1$'s. Suppose you have $n \geq 1$ commuting matrices $A_i$ over some finite field and such that $\sum_i A_i = ...
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Why do the cyclotomic cosets have the same size?

So I have this problem: when $p=2$ and $n=2^m-1$ show that $|C_1|=|C_3|=m$ where $m\geq 3$. $C_s=\{s,ps,p^2s,\dots p^{m_s-1}\}$ is the cyclotomic coset. I know that the size of the coset is $m_s$ ...
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To Factorize $x^{27}-x$ over $\mathbb F_3$.

Problem 7.5 in Chapter 15 of Artin's Algebra asks to factorize $x^{27}-x$ over $\mathbb F_3$. Here is what I have done. $x^{27}-x=x(x^{26}-1)= x(x^{13}-1)(x^{13}+1)$. In am having trouble ...
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61 views

Proving that $X^4+1$ is reducible over all finite prime fields

In that article, I prove that the polynomial $X^4+1$ is reducible over all finite prime fields of odd characteristic. The proof is based on the fact that for $p$ odd prime, the multiplicative group ...
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Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?

How to show that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal? Is it possible to find the cardinality of $\mathbb{Z}[i]/I$ as well? I know how to show that it is an integral ...
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Zero as a repeated permental root for a matrix over a finite field

All, Suppose $A \in Mat(n, \mathbb{F}_{q})$ for $q$ prime, $q \geq 5$, and $n \geq 2^{q-2}$ . Let $\pi_A(x)$ be the permanental polynomial for $A$. That is, \begin{equation*} \pi_{A}(x)=per(xI-A). ...
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Counting couples of square-free polynomials over finite fields

I have a curve defined by the following equations over the finite field $\mathbb{F}_q$ with $q=p^r$ with $p \geq 3$: $$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\ y_2^2=h_2(t) ...
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Automorphism group of the general affine group of the affine line over a finite field?

I am wondering what the structure of the automorphism group of the general affine group of the affine line over a finite field looks like. I'll make that a bit more precise: If $k$ is a finite field, ...
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What is known about the numbers $M_p = \left\vert C(\mathbb{F}_p )\right\vert$?

There is a question (2.4.c) marked ** (to denote "extremely difficult/currently open problem") in Silverman and Tate's Rational Points on Elliptic Curves which I found really interesting and wondered ...
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In cryptography, why do we reduce elliptic curves over finite fields?

What's wrong with real numbers? Is the continuous logarithm problem "easy" to solve for elliptic curves? Here's what I believe: elliptic curves over the real numbers have infinitely many points, many ...
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Is $K\mathbb P^n\cong K^n\mathbin{\dot\cup}K\mathbb P^{n-1}$ for all fields $K$?

I know that for $K=\mathbb R$, the statement $\mathbb R\mathbb P^n\cong\mathbb R^n\mathbin{\dot\cup}\mathbb R\mathbb P^{n-1}$ (where $\cong$ denotes set isomorphism) holds. Is the identity $$K\mathbb ...
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In a finite field, is there ever a homomorphism from the additive group to the multiplicative group?

I know that in infinite fields, such as $\mathbb{C}$, the mapping $e^x$ is a homomorphism from the additive group to the multiplicative group, and I was just wondering if in any finite field, there ...
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Prove that $x^{p^n} -x +1$ is irreducible over $\mathbb{F_p}$ only when n=1 or n=p=2

Question:Prove that $x^{p^n} -x +1$ is irreducible over $\mathbb{F_p}$ only when n=1 or n=p=2 I know it is a duplicate question. However, someone gave some nice hints on this problem and I want to ...
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Mapping the additive group of a finite field of order $2^n$ to its multiplicative group

In a finite field $F$ of order $2^n$, we know that its additive group is isomorphic to $(\mathbb{Z}_2)^n$. We also know that $(\mathbb{Z}_2)^n$ can be thought of as the set of all $n$-digit binary ...
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Binary Goppa Codes: Calculating code characteristics (traces, length L, distance d, k)

I am having (many) troubles with binary Goppa Codes. My question is at the moment: How do I calculate the trace on given points $tr(\alpha^u)$? For example: Given a finite field $GF(2^6)$ and the ...
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Showing that the fixed points of a homomorphism form a finite field.

I have the following question on my problem set. Suppose that $\textsf{k}$ is a field and $\phi:\textsf{k}\to\textsf{k}$ is a homomorphism. Check that $\textsf{k}^\phi=\{x\in\textsf{k}:\phi(x)=x\}$ ...
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Irreducible Polynomials $GF(2^4)$: Why is $x^4 + x^2 + 1$ reducible?

I am currently working with $GF(2)$, in particular with $GF(2^4)$. One task is to find all irreducible polynomials. I have found ways of reducing the list of all candidates drastically. In my current ...
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Boolean Polynomial Mutliplication modulo an irreducible polynomial

I am currently reading a paper on the Mathematics of RAID 6 by Peter Anvin and cannot get my head around the notation or results used describing multiplication by {02} (hexadecimal). Here is how he ...
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Distribution of Trace values

I try to prove that ${2^{n-1}}$ elements of the field $\mathbf{F}_{2^{n}}$ have a Trace with value 1, while the other ${2^{n-1}}$ elements have a Trace with value 0. I started to show that Trace(1) ...
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62 views

Irreducibility of polynomials over finite field

If I want to show that a certain polynomial is irreducible over a finite field, which methods do I have? In particular how can I show that $X^4-3$ is irreducible over $\mathbb F_5$ The idea which I ...
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resultants over finite fields vs resultants over their closure

I'm having a little trouble grokking the following line of argument. I have two monic, univariate polynomials $p(x)$ and $q(x)$ over $\mathbb{Z}/p\mathbb{Z}$ that share a common root, where $p$ is ...