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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Trace over $\mathbf{F}_2$ of $1/(\alpha + \alpha^{-1})$ where $\alpha^{2^n+1} = 1$

Let $n$ be an integer $\ge 2$. Put $K = \mathbf{F}_{2^n}$ and $L = \mathbf{F}_{2^{2n}}$. Let $\alpha$ be an element of $L$ such that $\alpha^{2^n+1} = 1$ and $\alpha \ne 1$. Using Sage, I have noticed ...
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Why is $E[l]\cong\mathbb Z/l\mathbb Z\times\mathbb Z/l\mathbb Z$ for an elliptic curve $E$?

René Schoof's 1995 paper contains the following statement about an elliptic curve $E$ (at the bottom of page 233): [...], we use the subgroup $E[l]$ of $l$-torsion points of $E(\overline{\mathbb ...
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Counting total number of monic irreducible polynomials of all degrees $k$ that divide $m$.

Why is the following relation counting monic irreducible polynomials of all degrees $d$ that divide $m$ true? \begin{equation} \sum_{d\ |\ m}\left(\frac{1}{d} \sum_{c\ |\ d} \mu(d/c)\ p^{c}\right) = ...
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Explicitly construct a field with 729 elements

I would like to construct a field with 729 elements. I know that $729 = 3^6$, and that I have to find an irreducible monic polynomial of degree 6 over $GF(3)$. I chose the polynomial $x^6 + 2x^2 + 1$, ...
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40 views

What happens to symplectic basis if bilinearity condition is weak

Let $B:V\times V \rightarrow K$ be a (weak!) bilinear form where $K$ is a finite field with base field $F$ and $V$ a vector space over $K$. Let $u,v \in V$ and $\lambda \in F (!)$ $B(u + v, w) = B(u, ...
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Let $c_0, \cdots, c_{n-1}$ be elements of $\mathbb{F_{2^k}}$, find the sum $\sum\limits_{0\leq i < j <n}c_ic_j$

Let $c_0, \cdots, c_{n-1}$ be elements of $\mathbb{F_{2^k}}$, find the sum $\sum\limits_{0\leq i < j <n}c_ic_j$. It is true that if $k=1$ and $d$ be the number of non-zero elements, then ...
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Elementary questions about polynomials and field extensions

Let $$f(x)=x^2+x+1.$$ This is irreducible in $\mathbb{Z_2}[x]$, and thus $\mathbb{Z_2}[x]/(f(x))$ is a field $K$ where $(f(x))$ is a principle ideal. I don't quite understand how I find that ...
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22 views

Prove $(F,+)$ isomorphic to $(\mathbb{Z}_{p}, +) \times (\mathbb{Z}_{p}, +) \times … \times (\mathbb{Z}_{p},+)$ ( $n$ times)

Let $F$ be a field of order $p^n$ for some prime $p$ and positive integer $n$, and let $\mathbb{Z}_{p} \subset F$ be a prime field of $F$. Prove the additive group of $F$, that is the group ...
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140 views

Can we rewrite $Tr(ab^2)$ as $Tr(f(a)b)$?

Can we rewrite $Tr(ab^2)$ as $Tr(f(a)b)$ where $Tr:F_{2^{kn}}\rightarrow F_{2^{k}} $ is trace map, $k \neq 1$, $f$ is a function just depends to $a$.
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Linear equations - how to find the solution over the boolean field closest to zero

I want to solve a system of linear equations over the field of $F_2$, in a way such that the solution vector is as close to the zero vector as possible. For example, suppose I have a system of ...
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1answer
48 views

When is the Frobenius the identity?

If $f$ is an irreducible polynomial of degree $n$ over $\mathbb{F}_{p}$, then $\mathbb{F}_{p}[x]/(f)$ is the finite field $\mathbb{F}_{p^{n}}$ and the map $a \mapsto a^{p}$ is the Frobenius ...
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44 views

Polynomials in Finite Field

Given $\Bbb F_{32}=\Bbb F_2[X]/(X^5+X^2+1)$ where the polynomial is irreducible over $\Bbb F_2$, how would I compute $(a^4+a^2)*(a^3+a+1)$ given $a=[X]$ is the congruence class of [X]? Multiplying ...
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56 views

Does this imply that $F_1$ and $F_2$ are isomorphic as fields?

Suppose $F_1$ and $F_2$ be two finite fields such that additive groups of $F_1$ and $F_2$ are isomorphic and also multiplicative groups of $F_1$ and $F_2$ are isomorphic.Does this imply that $F_1$ ...
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35 views

On diagonizability of commutating matrices

Let $A$ and $B$ be $n\times n$ matrices over the Galois Field of order $p$ ($p$ is a prime). Suppose that $A$ and $B$ are diagonizable matrices and that they commutate. Is it possible to make them ...
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35 views

How to guarantee existence of a finite field [duplicate]

Existence of a finite field: Solution: I can understand that if I have a finite field $F$ of characteristic $p$ where $p$ is prime then I can consider $\mathbb Z_p$ as its prime field and hence $F$ ...
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1answer
57 views

How to do multiplication in $GF(2^8)$?

I am taking an Internet Security Class and we received some practice problems and answers, but I do not know how to do these problems , an explanation would be greatly appreciated Try to compute the ...
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1answer
88 views

Irreducibility of $1+x+\dots+x^{n-1}$ over $\mathbb{F}_2[x]$

Can someone provide a reference of the proof (or the proof itself) of this statement? The polynomial $1+x+\dots+x^{n-1}$ is irreducible over $\mathbf{F}_2[x]$ if and only if $n$ is an odd prime and ...
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69 views

Matrix which is not similar to it's transposed

Let $V$ be vector space over a field $\mathbb{k}$. I can prove that any matrix is similar to its matrix transpose if $\mathbb{k}$ is an infinite field, but is this still true when $\Bbb k$ is finite? ...
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1answer
41 views

Do non-graded rings exist?

I've been reading about graded rings recently, and I was wondering if there exist commutative or non-commutative rings for which no non-trivial gradings exist? That is, a ring for which if $R ...
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30 views

show that $K=\Bbb F_{q^m}$, where m is the order of $q$ in the group of units $\Bbb {Z}_n^*$ of the ring $\Bbb Z_n$.

Let $q$ be power of a prime $p$, and let $n$ be a positive integer not divisible by $p$. We let $\Bbb F_q$ be the unique upto isomorphism finite field of $q$ elements. If $K$ is the splitting field of ...
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33 views

How to deduce the irreducible factors of $x^4 +1$ in $\Bbb F_3$.

I have two questions: 1) How to deduce the irreducible factors of $x^4 +1$ in $\Bbb F_3$. Clearly the irreducible factors will be of degree $2$. But can anyone calculate it for me? 2) I have proved ...
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Finding units in cyclotomic fields

I want to classify the six units in $\mathbb Z[\zeta_3]$, where $\zeta_3$ is a primitive cube root of unity. I know the basic idea of this is to show that the norm of $\alpha \in \mathbb Z[\zeta_3]$ ...
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38 views

Finding the minimal polynomial and its conjugates without a matrix

Let $K=\mathbb Q\left(^3\sqrt{5}\right)$ and $\alpha=a+b\left(^3\sqrt{5}\right)+c\left(^3\sqrt{5}\right)^2$. How do I find the minimal polynomial $f_\alpha$ of $\alpha$ over $\mathbb Q$? I am already ...
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Finite field question involving the trace and a permutation.

Let $q$ be a power of a prime $p$, and $m,l$ positive integers with gcd$(l,q^m-1)=1$. Denote $Tr$ to be the trace of $GF(q^m)$ over $GF(q)$. Suppose that there exists a nonzero $\gamma \in GF(q^m)$ ...
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14 views

Show that $f(c)=0\forall c\in GF(p),deg(f)<p\Leftrightarrow f(X)=0$; $f(X)\in GF(p)[X]$

Let $F = GF(p)$, where $p$ is a prime integer, and let $g$ be an arbitrary function from $F$ to itself. Show that there exists a polynomial $f(X) ∈ F[X]$ of degree less than $p$ satisfying the ...
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Frobenius injective for finite fields - what about $\mathbb{F_{p^n}}$

Quick question about the Frobenius endomorphism. My lecture notes and wikipedia say that the Frobenius is injective for finite fields. However, if we look at $\mathbb{F_4}$, we have $$\text{Frob}(2) = ...
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1answer
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Finite matrix power over $\Bbb F_q$

What is largest $s\in\Bbb N$ such that a matrix $M\in\Bbb F_q^{n\times n}=\Bbb F_{p^r}^{n\times n}$ could satisfy $$M^i\neq I,\quad\forall i\in\Bbb Z_+:0<i<s$$ $$M^0=M^s=I?$$
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Artin-Chevalley Theorem.

Let $p$ be a prime, $q=p^n$, and $\mathbf F_q$ denote the finite field with $q$ elements. Problem 7 in Section 2.12 of Basic Algebra Vol. 1 by N. Jacobson asks the following: Let ...
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Primes $p$ such that $ord_p(3)$ is a power of $2$

I'm trying to solve this problem, and i need to find all primes $p$ for which $ord_p(3)$ is a power of $2$. If such primes exist then they are of the form $p=2^km+1$ with $k\geq1$ and $m$ an odd ...
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Polynomial Evaluation

My question is to some extent related to cryptography, but I'd like the mathematicians answer my question, please (as their answers are usualy more clearer than cryptographers). Consider I have a ...
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86 views

Matrices over a finite field with given Jordan normal form over the algebraic closure

Can one describe the (conjugacy classes of) square matrices over a finite field such that over the algebraic closure of this finite field their Jordan normal form consists of one Jordan block? (Such ...
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Square roots of $\Bbb F_p$

Can anyone please help me to show that $\Bbb F_{p^2}$ contains all the square roots of $\Bbb F_p$ where $p$ is a prime? Thanks for any help.
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1answer
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Squaring Polynomial over $\Bbb F_2[X]$ Is Equivalent to Squaring Argument

Thanks to some assistance below, I can now show that if $g(X) \in \Bbb F_2[X]$ then $g(X)^2 = g(X^2)$. Is there some more direct way to prove this special case (not that the original proof is ...
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Prove that if $I-A$ is invertible then $I-A^p$ is invertible in $\mathrm{Mat}_n(\mathbb{Z}_p)$.

How do you prove that if $I-A$ invertible then $I-A^p$ is invertible for $A \in \mathrm{Mat}_n(\mathbb{Z}_p)$, where $p$ is prime?
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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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1answer
32 views

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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52 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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128 views

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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29 views

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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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 ...
3
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1answer
50 views

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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1answer
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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38 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 ...