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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106 views

Trace of Frobenius of elliptic curve is integer

I recently started to read the book "Arithmetic of Elliptic Curves" by Silverman. And I can't solve an exercise 5.10. Let $E/\mathbb F_q$ be an elliptic curve and $\phi$ is Frobenius endomorphism, ...
4
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1answer
49 views

Multiplicative order in field extension

Let $F/K$ be some field extension (both are finite fields) and $u$ be some element in $F$. I want to know if $u^{|K|} = u$ implies $u \in K$. And why?
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1answer
23 views

Finite Field Homomorphism

Let's say that I have a finite field K with characteristic 2. I define @ as a map where @ : K -> K, and x -> $x^2$. First of all, what are some examples of fields like K? I initially thought it ...
2
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1answer
103 views

Algebraic problem - irreducible polynomials

Let $p$ be a prime number and $q=p^n$ with $n \in \mathbb{N}$. We define the polynomial$$ F_m:=\prod \left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, nomic}, \text{deg}(f)=m \right\rbrace $$ ...
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41 views

set of integer modulo m for composite values

Is there a Set of integer modulo m, Zm, where m is composite and the set is a field, (all of its elements having a multiplicative inverse) ? I have heard that the set of integer modulo 4 cannot be a ...
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1answer
78 views

Order of element of multiplicative group of finite field mod polynomial

If $K$ is a finite field of size $q$ and $f$ is a degree $n$ polynomial in $K[x]$, then we can form the quotient field by modding out this polynomial. Elements of this quotient field are of the form ...
3
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1answer
39 views

Simplifying presentation of elements of finite field

Let me describe my question through an example. Finite field of order (for example) 8 can be constructed as $\mathbb{F}_8 = \mathbb{F}_2[t]/(t^3 + t + 1)$. So one of a natural presentation of the ...
2
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1answer
23 views

Decision Diffie Hellman in finite fields

Is there an efficient mathematical algorithm for Decision Diffie-Hellman problem in a finite field $F_q$? I have found a detailed analysis of many more involved or specific cases but nothing on the ...
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2answers
74 views

When does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$?

Let $p$ be a prime integer, and let $q=p^r$ and $q'=p^k$. For which values of $r$ and $k$ does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$? From Artin's Algebra, Chapter 15, problem 7.12 from the ...
3
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2answers
87 views

Finite field with algebraic element over $\mathbb{Z}_p$

Let $F$ be a finite field of characteristic $p$. Show that every element of $F$ is algebraic over $\mathbb{Z}_p$. Since char$(F)$ = $p$, $\forall a \in F $ not zero, $ap$ = 0. We also have the same ...
5
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2answers
173 views

Understanding Intel's white paper algorithm for multiplication in $\text{GF}(2^n)$?

I'm reading this Intel white paper on carry-less multiplication. For now, suppose I want to do multiplication in $\text{GF}(2^4)$. We are using the "usual" bitstring representation of polynomials ...
3
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2answers
79 views

Group-Isomorphism problem

I want to find an group-isomorphism $$ \psi : (\mathbb{Z}/8\mathbb{Z},+) \longrightarrow \mathbb{F}_9^\times $$ which should be used to multiply elements in $\mathbb{F}_9$ or to find the inverse ...
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3answers
33 views

Linear Independence for different fields

I have a statement for a space over $R^n$: {x, y, z} is linearly ind. $\implies$ {x + y, x + z, y + z} is linearly independent Quick proof: a(x+y) + b(x+z) + c(y+z) = 0 $\implies$ (a+b)x + ...
5
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0answers
57 views

The reducibility of polynomial $x^{2^n}+x+1$ over $Z_2$ [duplicate]

I meet the problem in Hungerford's Algebra, page 282 as follows: If $n>2$, then the polynomial $x^{2^n}+x+1$ is reducible over $Z_2$ . I have no any idea on this. But I really want to know how to ...
2
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1answer
69 views

Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $ (\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
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4answers
86 views

A finite sum of $1$ equals $0$ in a field with finitely many elements.

I need to prove that for field $\mathbb{F}$ with finitely many elements $\exists n \in \mathbb{N}$, $\underbrace{1+1+1+...+1}_{n \text{ times }} = 0$. I can see why this is true, since there is a ...
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1answer
61 views

The degree of an irreducible polynomial divides an integer n.

Let $F$ be a finite field such that $|F|=q$, and $f(x)\in{F[x]}$ is irreducible with $deg(f)=d$, $g(x)=x^{{q}^{n}}-x$. Then prove that $f$ divides $g$ if and only if $d$ divides $n$.
2
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3answers
65 views

The Frobenius Automorphism

If a finite field $F$ has characteristic $p$, prove that every element $a\in F$ can be expressed as $a=b^{p}$ for some $b\in F$? Hint: Frobenius Automorphism. Isn't the Frob Automorphism about $a$? ...
3
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1answer
38 views

Find irreducible polynomials in $\mathbb{Z}_{3} [x]$ having as roots $a + 1$ and $a - 1$

$x^{2} + 1$ is irreducible in $\mathbb{Z}_{3} [x]$, and so K = $\mathbb{Z}_{3} [x] / \langle x^{2}+ 1\rangle$ is a field with 9 elements. Let $a$ in K be a root of $f(x)$. Find irreducible polynomials ...
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2answers
46 views

Finding a gcd in the form of a monic polynomial

The question is to find the greatest common divisor (in the form of monic polynomial) in $\Bbb F_5[X]$ of $f=x^2-x+4$ and $g=x^3+2x^2+3x+2$ I used the Euclidian Algorithm for polynomials and found ...
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0answers
32 views

permutation polynomial

If we have GF(4) as an extension field, we can define a permutation polynomial of GF(4) like L(x), a linearized polynomial, of the followinf form: L(x)= \sum_{s=0}^{\r-1} a_s x^(q^r)e Is it possible ...
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1answer
44 views

basis for finite fields as a vector space

If we consider GF(4) as a vector space over GF(2), the basis of GF(4) includes two elements 1 and a. Due to this fact that for an arbitrary vector space we can find several basis, what are other bases ...
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77 views

Multivariable irreducible polynomials over finite fields

It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it. For any $f(x_1,\dots, x_n)=\sum ...
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1answer
51 views

Solutions of $x^d=1$ in a finite field

Let's consider the polynomial $x^d-1$. Theory tells us that it can have at most $d$ roots in (any extension of) a given field. Here's my problem: let $A$ be the vector space spanned by ...
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0answers
39 views

Solve equations over field of size 2

I am trying to solve this problem: Let $\alpha_1x_1 + \alpha_2 x_2 + \alpha_3x_3 + \alpha_4x_4 + \alpha_5 = 0$ and $x_1x_2+x_3x_4=0$ be equations over field of size $2$. Show that we can't ...
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1answer
35 views

When is the Frobenius endomorphims an isomorphism?

I did this problem, but now I'm left with more questions! Suppose $f(x)$ is a monic irreducible polynomial of degree $3$ over $GF(2)$. Prove that if $a$ is a root of $f$ in an extension of $GF(2)$, ...
5
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1answer
71 views

Given a finite field $\mathbb{F}$ of order $2^n$, how to construct a field of order $2^{2n}$?

Specifically, I would like to construct a field of order $2^{2n}$ with elements being $2\times2$ matrices whose entries are elements of $\mathbb{F}$. I know the complex numbers can be represented as ...
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35 views

Reed-Solomon encoding in GF2?

I'm vaguely familiar with Reed-Solomon encoding + know that it's generally done in GF(256). Is there any way to use GF(2)? If I have a hardware LFSR (as is used for CRC calculation), can I make use ...
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2answers
22 views

Can a set containing a single vector from a vector space over a finite field be linearly dependent?

Take the set $S=\{v=(1,1)\}\subset F_2 ^2$. $v+v=(0,0)$ is a linear combination of vectors from $S$. Is $S$ linearly dependent?
3
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0answers
50 views

How to prove how many ireducible polynomials are in a polynomial ring over a finite field.

Can someone help me out with some problem concerning the cardinaliy of ireducible polynomials in $(\mathbb Z/p\mathbb Z)[x]$? Here first why the problem appeared, I want to prove that ...
6
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3answers
250 views

Polynomials over finite fields

I’ve come across this problem in a coding theory course, and neither I nor several of my colleagues could solve it to our satisfaction. Let $F:=\mathrm{GF}\left(q\right)$ denote the field with $q$ ...
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1answer
21 views

Prove that the image $\alpha$ of $X$ in $\left(\mathbb{F}_3[X]/(X^3-X^2+1)\right)^{\times}$ is a generator

I'm trying to do an exercise of my homework that sais I have to prove that the iamge of $X$ in $K^{\times}=\left(\mathbb{F}_3[X]/(X^3-X^2+1)\right)^{\times}$ is a generator. Acording to what I know, ...
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0answers
29 views

Action of $\mathbb{F}_{p^2}^\times/\mathbb{F}_{p}^\times$ on $P^1(\mathbb{F}_p)$

Let $p$ be prime. Let $\alpha$ be a generator of the finite field $\mathbb{F}_{p^2}$. So, $\mathbb{F}_{p^2}=\mathbb{F}_p[\alpha]$. Multiplication by $\alpha$ is an $\mathbb{F}_p$-linear operator on ...
0
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1answer
24 views

Minimum polynom of an element in $K=\mathbb{F}_5[x]/(x^2-2)$

I want to know how to calculate the minimum polynom of an element $\alpha$ in $K=\mathbb{F}_5[X]/(X^2-2)$ where $\alpha$ is the image on $K$ of $X+2$ I'm already verficated that $K$ is a field. As I ...
3
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1answer
68 views

How to find multiplicative orders of all elements in field $\Bbb F$ (say $\Bbb F_{13}$)?

I am working on some finite fields and I was referring to some online class material. Is there any way to find the multiplicative orders of all elements in a field $\Bbb F$?
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18 views

Find maximum rank vector

I have a $n\times n$ matrix with coefficients from a finite field $F$. For any vector $v$ of $F^n$, I consider the sequence: $v_1=v$ and $v_{n+1}=M.v_n$ The rank of $v$ is the rank of the sequence ...
3
votes
0answers
62 views

Set with roots of $x^{p^{q}}-x$ is $\mathrm{GF}(p^q)$?

If $\mathrm{GF}(2^6) = \mathbb{Z}_2 [x]/ (x^6 + x + 1)$ and $u$ is a generator with $o(u)= 63$ how do I show $\mathrm{GF}(2^3)=\{0,1, u^9, u^{18}, u^{27}, u^{36}, u^{45}, u^{54}\}$? Here's what I ...
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2answers
48 views

Finding the kernel and image of a linear transformation over the field $\Bbb Z_2$

Given the A vector space $V$ over the field $\Bbb Z_2$ and a linear map $t:V \to V $ .following matrix $T =\begin{bmatrix}3& -1 &1\\-1 & 5 & -1\\ 1 & -1 & 3\end{bmatrix}$ ...
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3answers
109 views

Homogenous polynomial over finite field having only trivial zero

Is there a way to construct homogenous polynomials of small degree over a certain finite field having only trivial zero? For instance, the polynomial $f (a, b, c) = a^3 + b^3 + c^3 - 3abc - 3a^2b - ...
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1answer
53 views

Galois Field $2^3$ Multiplication of $111$ and $111$

In Galois Field Multiplication is performed as a binary multiplication between the two binary values representing polynomials and then is reduced by a reduction polynomial So if i have the binary ...
2
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3answers
49 views

A hint on why if $c$ is not a square in $\mathbf{F}_p$, then $c^{(p - 1)/2} \equiv -1 \mod p$

Let $\mathbf{F}_p$ be a finite field and let $c \in (\mathbf{Z}/p)^\times$. If $x^2 = c$ does not have a solution in $\mathbf{F}_p$, then $c^\frac{p - 1}{2} \equiv -1 \mod p$. I will try to prove the ...
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1answer
45 views

Size of field extension is at most $p^d$

I have a field extension $\mathbb{F}_p(\alpha)$ where $\alpha$ is a root of the irreducible polynomial $f \in \mathbb{F}_p[t]$ and I know that $\alpha ^{p^d} = \alpha$, where $p$ is some prime. I'm ...
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1answer
65 views

$\exists a \in \mathbb{F}_{11}$ such that $\mathbb{F}_{11}[x]/\langle x^5-a\rangle$ is a field

Prove that there exists an element $a \in \mathbb{F}_{11}$ such that the quotient ring $\mathbb{F}_{11}[x]/\langle x^5-a\rangle$ is a field. I wrote that it is equivalent to showing that there is ...
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2answers
45 views

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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1answer
45 views

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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1answer
36 views

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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0answers
51 views

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, ...
3
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1answer
122 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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0answers
43 views

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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2answers
43 views

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 ...