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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Algebra: Field Theory Question

I am being asked to find the following: Let $F$ denote the field $\dfrac{\mathbb{F}_2[\alpha]}{(\alpha^3 + \alpha + 1)}$. Simplify $\alpha(\alpha + 1)(\alpha + 1)$ in $F$ and calculate ...
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39 views

Generating elements of large group of units of a field.

Suppose $F$ is a large but finite field, and $F^\text{x}$ is the group of non-zero elements of the field. It is known that this group is always cyclic, and we could ask a natural question "what are ...
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50 views

Orthogonal matrices over GF(2^t) with the first column fixed

I'm interested in the existence of certain orthogonal transformations over $\mathbb F = \mathrm{GF}(2^t)$: matrices $M: \mathbb F^d \to \mathbb F^d$ for which $M^{\!\!\;\mathsf T\!\!\;} M = I_d$ over ...
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What is the best known lower bound for: $\max_{2\leq i\leq p-1}(ord_n(i)) $?

Given an in integer $n$, and let $p$ be its smallest prime divisor (you can assume that $p$ is very large ). Let $ord_n(i)$ denotes the order of $i$ as an element of $\Bbb Z_n^*$ the multiplicative ...
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Greatest common divisor of polynomial in Finite Field(256), AES

Have assigment and will use it as example, found solution computationaly, want to understand idea. It is about SubBytes procedure in AES, particulary about finding inverse of polynomial. Suppose we ...
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81 views

Equivalence of $\mathbb{F}_p$ and $\mathbb{Z}/p\mathbb{Z}$

I feel like I have some fundamental misunderstanding and I'm not really sure how to phrase this question, but here's a first attempt. In Child's Concrete Introduction to Higher Algebra (ISBN: ...
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105 views

Suppose that $a$ and $b$ belong to a field of order $8$ and that $a^2 + ab + b^2 =0$ then $a=0$ and $b=0$ . [duplicate]

Suppose that $a$ and $b$ belong to a field of order $8$ and $a^2 + ab + b^2 =0$. Then $a=0$ and $b=0$. Do the same when the field has order $2^n$ with $n$ odd? If one of the term is zero, i.e. ...
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I have to show this polynomial is irreducible. [closed]

Suppose that $p(x)=x^9+x^8+x^4+x^2+1 \in \mathbb{Z}_2[x]$. I have to show this polynomial is irreducible.
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Determine with proof the number of monic irreducible polynomials of prime degree $p$ over $F$

Let $F$ be a field with $|F|=q$. Determine with proof the number of monic irreducible polynomials of prime degree $p$ over $F$, where $p$ need not be the characteristic of $F$. I know that ...
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41 views

Finding normal basis for GF(q^m) over GF(q)

Could you kindly explain, how can one find a normal basis for GF$(3^6)$ over the GF$(3^2)$? As I understood, I should start with finding the polynomial in a form $$a(x^2) + (a^9)x + a^{81},$$ which ...
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Shortened Reed-Solomon proving p(D) is primitive

Assume we have a shortened $(n=18, k=12, t=3)$ Reed Solomon code in $GF(2^{8})$.Let $\alpha$ be a primitive element of $GF(2^{8})$. Consider the primitive polynomial given by: $p(D) = D^{8} + D^{4} + ...
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1answer
30 views

Why are the elements of GF(q), whose characteristic is 2, all squares? [duplicate]

If $a\in GF(2^n)$, then there is the element in $GF(2^n)$ such that $x^2 = a$. Why?
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33 views

Is there any definition of such semi-bilinear?

$K$ is a finite field which not equal to its base field $F_2$. Let $f: V \rightarrow K$ be a function and $B(x,y)=f(x+y)+f(x)+f(y)$ such that $B(x+y,z)=B(x,z)+B(y,z)$ and $B(z,x+y)=B(z,x)+B(z,y)$ for ...
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38 views

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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37 views

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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71 views

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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43 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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43 views

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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23 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
51 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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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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38 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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60 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
93 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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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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56 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 ...
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31 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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42 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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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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50 views

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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42 views

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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43 views

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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101 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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2answers
55 views

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
21 views

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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19 views

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 ...
2
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1answer
36 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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30 views

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 ...