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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Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

Iā€™m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...
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Example for a corollary in finite field theory

I'm preparing a seminar and the problem is that I never had a lecture before in finite fields. So I had to learn everything by myself, and that is unfortunately not easy at all... Can anyone help me ...
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Having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space over $\mathbb{F}_p$.

As the title states, I'm having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space V over $\mathbb{F}_p$. Clearly it is dimension $n$. How can we work with this vector space? ...
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Show that a map is not an automorphism in an infinite field

How should I show that a map $f(x) = x^{-1}$ for $x \neq 0$ and $f(0) = 0$ is not an automorphism for an infinite field? Thanks for any hints. Kuba
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44 views

The characteristic of the field $GF(p^n)$

How to show that characteristic of the field $GF(p^n)$ is $p$? I have come across this fact on Wikipedia webpage, but don't know how to prove it. Thanks
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When Frobenius map equal to multiplication-by-m map

Here is a homework, the result brought me some trouble. Let $p = 7$, and consider the finite field ${ \mathbb{F}}_{p^{2}}$ , which we may represent explicitly as $${ \mathbb{F}}_{p^{2}}\simeq ...
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61 views

an Example of Elliptic Curve over finite field has no CM

I have known this property (from Silverman's The Arithmetic of Elliptic Curves): Let $\operatorname{char}(K)=p>0,$ and let $E/K$ be an elliptic curve with $j(E)~ \overline{\in}~ \overline{ ...
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28 views

What does it mean to say that an element 'satisfies' a polynomial?

In the context of finite fields, the definition of a primitive element $\alpha$ is given by: $\alpha$ is primitive if it generates all elements of $F_q - \{0\}$ when raised to powers up to $q-1$. ...
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55 views

Galois Group of $x^4+x-1$ over $\mathbb{F}_3$

Consider the finite field $\mathbb{F}_3$ and define the polynomial $f(x)=x^4+x-1$ over $\mathbb{F}_3$. I want to find its Galois Group. I observe that $f$ has no root over $\mathbb{F}_3$, so if it ...
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25 views

matrix for frobenius map of finite fields

I would be thankful if you could help me : I have studied many things about Galois fields, but now I am not sure about my understanding of frobenius maps. For example can anyone help me the matirx of ...
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31 views

Canonical representation of finite field

If there a canonical representation of finite fields $\Bbb F_{p^n}$ for $n>1$? By canonical I mean that if I were to say to someone else "this bunch of bits represents an element of $\Bbb ...
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56 views

Roots of Artin-Schreier equation

Let $a \in \mathbb F_q, q=p^f$. Is it true that $x^p-x-a$ has a root in $\mathbb F_q$ iff $tr_{\mathbb F_q/\mathbb F_p}a=0$?
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Discrete Logarithm Problem in $GF(p^m)$

I have question regarding DLP in $GF(p^m)$ I know the algorithms for solving the DLP in $GF(p)$ like Baby Step-Giant Step, Pohlig-Hellman etc... But what if we move into the $GF(p^m)$ and are ...
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37 views

Non-obvious deduction regarding conjugates in $\text{GL}_2(\mathbb{F}_p)$

Let $\text{GL}_2(\mathbb{F}_p)$ act on $\mathbb{F}_p^2,$ the set of $2$-vectors with entries in $\mathbb{F}_p$, by matrix multiplication. $[\,$Prove that$\,]$ for any ...
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22 views

meanings determinants of matrices in finite field

Let's $\Bbb{Z}_q$ is finite field. ($q$ is prime number). Lets $A_1$ ā€“ set of matrices $n\times n$, such that $\det(M) = 1$, for any matrix $M \in A_1, A_2$ ā€“ set of matrices $n\times n$, such that ...
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How and in what context are polynomials considered equal? [duplicate]

There's two notions of equivalent polynomials floating around, one saying that $f = g$ iff they're equivalent as maps, and the other saying $f = g$ iff they're equal on each coefficient when written ...
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Prove that if $a^2 + ab + b^2 = 0$ then $a = b = 0$?

We are given that $a, b \in F_{2^n}$ where $n$ is an odd +ve integer. Suppose $a^2 + ab + b^2 = 0$ then we have either $a = 2^n-b^2$ or $a+b = 2^n - b^2$. Which implies that $\sqrt{2^n -a} = +-b $ or ...
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90 views

Prove that every extension of a finite field is normal

In book by Roman 'Field Theory' it is written that it is straightforward that every extension of a finite field is normal. However I just cannot see it. Can you help me with this problem? Thank
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Solving the discrete logarithm using index calculus, finite fields and factor bases.

(a) Let $p$ be the prime 1073741827, with $\Bbb{F}_p$ the corresponding finite field. A primitive root in $\Bbb{F}_p$ is equal to $g=2$. Use a factor base of primes up to 13 to find the discrete ...
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Can anyone explain the difference between a ring, a group, and a field in a way so that your average 15 year old can understand? [duplicate]

I can not understand key difference between them, because all on the web uses mathematical notations...Can anyone explain in easy way?...please.Thanks in advance.
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Isomorphism between finite fields adjoining a root

Let $p(x)=x^3+x^2+1$ and $q(x)=x^3+x+1$ be polynomials over the field $\mathbb{Z}_2$. Let $\alpha$ be a root of $p(x)$ and $\beta$ be a root of $q(x)$. Now let $K=\mathbb{Z}_2(\alpha)$ and ...
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How is the table generated for Galois Field?

If I want to generate tables for $01AB\quad 01AB$ for both addition and multiplication, how will it be generated? I am basically confused from this wikipedia example! I hope someone can clear it up ...
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Structure of a module and finite fields

I am trying the following problem which appeared on an exam for a grad course: Determine the structure of finite field $F$ as a module over $F_p[x]$ when $xv=Lv$. Here $L(z)=z^p$ and ...
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Existence of root of a polynomial over $\mathbb F_p$.

I came accross the following question and I can't find an easy proof of this fact : Let $p\geq 17$ be a prime number such that $p\equiv 1 \pmod 4$. Show that for any $z\in \mathbb ...
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1answer
57 views

Choosing a polynomial for CRC

CRC checksum is a homomorphism from polynomials over $\mathbb F_2$ to itself. As I understand, the map $f\mapsto g$ it is simply remainder from division $f$ by $p$, where $p$ is a fixed polynomial for ...
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32 views

Ring theory Algebra

Let F be a field of $8$ elements and A=set of all x belongs to F such that $x^7=1$ and $x^k \neq 1$ for all $k < 7$. then the number of elements in A is
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About Degree of Polynomial

Consider a set $M_{m\times m}(F)$ of matrices of order $m\times n$ over the finite field $\mathbb{F_p}$. The set $M_{m\times m}(\mathbb{F_p})$ forms a $Ring$ under the binary operations (addition, ...
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53 views

How many different proper subfields does $K$ have, where $K$ is a field of order$p^n$?

If $p$ is a prime, and $d$ is an integer greater than or equal to 1. Let $n= p^d$, then there exists $K$ being a field of order $p^n$. But how many different proper subfields does $K$ have? The ...
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Irreducible Polynomials over a Finite Field

I am motivated by this question, and want to find a solution to the following problem. Question: How many monic, irreducible polynomials of degree $n$ are there over the finite field ...
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Creating a matrix such that all the sub-matrices are max rank

Let $A\odot B$ denote the elementwise multiplication of matrices $A$ and $B$. Given a binary matrix $B_{m \times n}=[b_{ij}]$, $b_{ij} \in \{0,1\}$, I want to find a matrix $A=[a_{ij}]$, $a_{ij}\in ...
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What is an algebraically closed field of characteristic $p$?

I suspect that this is a very simple question, but I need to ask. My question is How do the fields of characteristic $p$ look like? If $K$ is a finite field of order $p^n$, then $K$ has ...
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Basis of finite field as vector space

If we consider GF(8) as a vector space over GF(2), what are the basis for GF(8)? and How can we define a dual space for GF(8) as a vector space?
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Determinant of a generalization of Moore matrices

The Moore matrix over $\mathbb{F}_q$ is the $n\times n$ matrix whose i'th row is: $a_i,a_i^q,a_i^{q^2},\dots,a_i^{q^{n-1}}$. The determinant of this matrix is the product of all linear combinations ...
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Question on finite fields

I was curious about this question: Let $p$ be a prime, and $d \geq 1$ and $K$ is a field of $p^d$. How many proper subfields does $K$ have? All I know if that a finite field has order $p^n$, where ...
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Can we calculate $a^{n i} \mod p$?

If we have a natural $n$ (not $0$), and a prime $p$, is it possible to calculate $$a^{n i} \mod p$$ where $i$ is the imaginary number $\sqrt{-1}$? SOME THOUGHTS Knowing that $a^{i \cdot i} = ...
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Degree of a Finite Field

Consider the finite field of characteristic $\mathbb{F}_{p}$ and the polynomial $f(x) = x^{p^{n}}$ - x. The splitting field $f(x)$ is a field $\mathbb{F}_{p^{n}}$ with $p^{n}$ elements. Given this ...
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How to give irreducible factorization in $\mathbb{Z}_5$?

I have this polynomial: $$f=x^5+2x^3+4x^2+x+4$$ How can i find the irreducible factorization(in $\mathbb{Z}_5$)?I can find the roots easily but thats not enough.
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Connecting inner products and the trace of a matrix

Hello, I'm currently trying to work through this question. However, I am struggling with understanding what I should do. I am aware of all of the definitions used throughout, however I cannot link ...
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What is the definition of a non-degenerate homogeneous quadratic form over a finite field?

I read in some finite geometry notes by S. Ball and Z. Weiner the following: A conic is a set of points of $PG(2,q)$ that are zeros of a non-degenerate homogeneous quadratic form (in $3$ ...
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26 views

Every irreducible polynomial in $\mathbb{F}_p[x]$ is separable?

How can I show this? I tried proving the contrapositive statement but didn't get anywhere. I think I may have to do something involving automorphisms of the splitting field of such a polynomial, and ...
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If $F$ is a finite field then there exists an irreducible polynomial in $F[x]$ with degree $n$ for all $n\in \mathbb{N}$.

If $F$ is a finite field then there exists an irreducible polynomial in $F[x]$ with degree $n$ for all $n\in \mathbb{N}$. How can I show this? A hint was given: 'Can you think of a condition that ...
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Why doesn't SAGE understand reduced expressions mod p in a finite field extension?

Suppose I have a finite field $\mathbb{F}_{13}$ and I would like to adjoin an element, $\zeta$, with order $3$, since $\mathbb{F}_{13}$ does not contain one. So consider $\mathbb{F}_{13}(\zeta)$. Then ...
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Isomorphism between finite fields

Refering to this question suppose I have $l(x):=x^3+x+1$ and $m(x):=x^3+x^2+1$. Then prove there is an isomorphism between $\mathbb{F}_3 [x]/l(x)$ and $\mathbb{F}_3[x]/m(x)$ I can say that elements ...
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Splitting Field of Irreducible Polynomial over a Finite Field

Suppose $f(x)$ is irreducible over $\mathbb{F}_p[X]$ and let $\alpha$ be a root of $f$ in some extension field. I want to prove the following claim. I have included thoughts below, but I am very ...
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Regarding doubt of order of element in a finite field

The problem goes as : Let $p$ be an odd prime & $\mathbb F_{p} =\mathbb Z/p\mathbb Z$.Show that: $x^{2}+1$ has a root in $\mathbb F_{p}$ iff $p \equiv 1 ( mod $ $4)$ . My Solution: $\mathbb ...
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Galois field theory

$\mathrm{GF}( 29^2)$ is created by adjoining the root of the irreducible quadratic $p= x^2 + 7x +15$ to the field $\mathrm{GF}(29)$ . The cubic polynomial $q = Y^3 + (26x + 26)Y^2 + (8x+22)Y +13x+23$ ...
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Multiplying in GF(128)

I know that in GF(128) $a + b = a \oplus b$. I have an multiplication table for GF(128). In this table $7\cdot 5 = 27$. How can I create table like this?
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My notes on $\Bbb{Z}/p\Bbb{Z}$-theoretic computational complexity

(Question at the very bottom) Def 1. Let $F = \Bbb{Z}_p$ be a finite field. Then an $F^k$-machine is a machine with $k$ input / output memory slots. All computations are done in the field $F$ and ...
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Subfields of irreducible polynomial fields with known dimensions

Let's say we have an irreducible polynomial, $h(x) = x^4 + x + 1 \in \Bbb F_2[x]$, and that L is a field equal to $\Bbb F_2 [x]/(h(x))$. How would I go about finding a subfield K such that $[L : K] = ...
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Do there exist polynomials not computable in polynomial time?

Motivation: Computing a problem in $k$ memory slots Do there exist polynomials in $R = \Bbb{Z}_p[z_1, \dots, z_k]$ that can't be computed in time polynomial in $k,p$? Thanks... Good luck! Edit. I ...