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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Show that the set {${\alpha \in F_{5^6} | F_5(\alpha)=F_{5^6} }$} contains 15480 elements

Question: Show that the set {${\alpha \in F_{5^6} | F_5(\alpha)=F_{5^6} }$} contains $15480$ elements Since this number is so large, I think there is some trick to get the answer. Also $15480$ is ...
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22 views

Finite geometry - how to determine parallel classes

I try to learn a little about finite geometry and I have now encountered the following exercise: Exercise: Construct the affine plane $\mathrm{AP}(\mathbb{Z}_3)$. Determine it's parallel classes ...
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In the finite field $F$ of characteristic $p$, is $a^{p^n} = a$?

If F is a finite field of characteristic $p$, $a$ is some element in $F$ and the number of elements in $F$ is $p^n$, is it true that $a^{p^n} = a$ for all $a$ in $F$? If it is, how could one prove or ...
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23 views

Are the following options correct in case of a field?

I am reading field theory and i can't answer the following: 1.Is $\Bbb R$ algebraic over $\Bbb Q$? 2.If a field is algebraically closed then it has characteristic as $0$. Obviously $[\Bbb R:\Bbb ...
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Amount of elements in finite vector space.

I'm trying to resolve two exercises from Kostrikin's book. Ex. 1 How many elements contains vector space $\mathbb{F}_p^n$ (vectors $(x_1,x_2,\dots,x_n)$ of length $n$) over a field ...
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33 views

Prove that polynomial doesn't have multiple roots over a splitting field

Let $p$ be a prime number, $m, s \in \mathbb{N}$ and $q = p^s$. Let $f(t) = \sum_{i=0}^m \lambda_i t^{p^{i}} \in \mathbb{F}_q[t]$. I'm asked to prove that $f(a\alpha+b\beta) = af(\alpha)+bf(\beta)$, ...
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25 views

find the number of distinct arrangements such that all columns are linearly independent

Can someone help me how to proceed with this question? An arrangement is made of $n^2$ scalars from $\Bbb{F}$ (a finite field) in $n$ rows and $n$ columns such that each row and column can be viewed ...
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32 views

Period of matrix-vector multiplication in vector space over $\mathbb{F}_{p}$

I want to verify following proposition and construct a formal proof whenever it is true. This question is related to this one. Suppose $\mathbf{A}$ is an $n \times n$ matrix over $\mathbb{F}_p$ such ...
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23 views

Prove that if $\mathbb{F}_{p^n} \subseteq \mathbb{F}_{p^m}$ then $n \mid m$

The order of $\mathbb{F}_{p^n}$ is $p^n$ and $\mathbb{F}_{p^m}$ is $p^m$. Since $\mathbb{F}_{p^m}$ is the largest field and it contains all the elements of $\mathbb{F}_{p^n}$ and the order of any ...
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check whether $\mathbb{Z}[x]/(x-3)$ is a field or not?

Find out whether Quotient ring $\mathbb{Z}[x]/(x-3)$ is a field? I know that $x-3$ is irreducible in $\mathbb{Z}[x]$ but further don't know.
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Upper bounds for the number of roots of polynomials, over finite fields, lying in given extensions

Let $F$ be the finite field with $q$ elements, where $q$ is a power of a prime, and let $E$ be its degree $n \geq 2$ extension. Let $f(x) \in F[x]$ such that $f(E) = F$. Clearly the number of distinct ...
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1answer
25 views

Counting elements of $\Bbb{Z}/2\Bbb{Z}(\alpha)$

I have the field $K=\Bbb{Z}/2\Bbb{Z}$, I proved that the polynomial $P(X)=X^3+X^2+1$ is irreducible. Then I know that the quotient $K[X]/P$ is a field of $8$ elements. Let now $\alpha$ be a root in ...
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33 views

Order of matrix and monic irreducible polynomial over finite field

I want to verify (and prove - in case it is true) the following proposition. Suppose $\mathbb{F}_p$ is a finite field and $m(x)$ is a monic irreducible polynomial over $\mathbb{F}_p$ with ...
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How to check that a cubic polynomial is irreducible?

I want to argue that argue that $\pi(\alpha)=\alpha^3+3\alpha+3$ is an irreducible polynomial over the finite field with 5 elements $\mathbb{F}_5$. My approach was just to check that $\pi$ has no ...
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62 views

Special subspace in vector space over $\mathbb F_5$

Let $\mathbb F_5=\{0,1,2,3,4\}$ is finite field of size $5$. I am trying to find minimal $n$ so that vector space of dimension $n$ over $\mathbb F_5$ contains $2$ linearly independent vectors so that ...
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44 views

Degree of a splitting field over finite field

Given the polynomial $f(x)=x^4+1$ in the field $\mathbb{F}_p$, prove that $[F: \mathbb{F}_p]\leq 2$, where $F$ is the splitting field of $f$. Suppose $a$ is one of the roots of $f$. Then (if ...
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25 views

Representation of elements in finite principal ideal local rings

I'm reading the book "Finite Commutative Rings and their Applications" by G. Bini and F. Flamini. In page 21, the authors state "The isomorphism $\mathbb Z_{p^n}/pZ_{p^n}\cong\mathbb Z_p$ justifies ...
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1answer
28 views

How to define $\mathrm{GF}(2)$ with elements $\{+1,-1\}$?

To explain how addition and multiplication works in Galois fields, almost all the resources use the example of smallest finite field $\mathrm{GF}(2)$ that has elements $\{0,1\}$. How can we define ...
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30 views

How many distinct roots within an algebraic closure

Let $E=\overline{F_2}$. How to find the number of distinct roots of $f(x)=x^{81}-1\in F_2[x]$ in $E$? So far as I tried, I factorised $f$ into $$f(x)=(x-1)(x^{80}+x^{79}+\cdots+x+1)=(x-1)g(x)$$ ...
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Generating irreducible polynomials over $GF(2)$

I just want a clear answer to generating all the irreducible polynomials upto a certain degree in GF(2) (in my case upto 16) I am storing the prime polynomials as unsigned int
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3answers
157 views

The elements in a finite field.

I am reading some lecture notes from MIT Open Courseware. One of the theorems states that the elements in a finite field of order $q$ are the $q$ distinct roots of the polynomial $x^q - x$. I can ...
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Raising to the pth power and using Vieta in finite fields with characteristic p

Any finite field $K$ with characteristic $p$ is isomorphic to $\mathbb{Z}_p[t]/(f)$ for some irreducible $f\in\mathbb{Z}_p[x]$. (From now on, assume $K=\mathbb{Z}_p[t]/(f)$.) Since ...
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Is the splitting field of $x^2+1$ over $\Bbb Z_2$ is $\{0,1,i,1+i\}$.

I am not sure about my answer. Since x^2+1 is reducible with respect to Z2 but X^2-1 is irreducible with respect to Z2 so I made this finite field {0,1,i,1+i} nd said it as the splitting field of ...
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3answers
73 views

What are the fields such that $x^4 = 1$ for every $x$ in the multiplicative group

What are the fields $K$ such that $x^4 = 1$ for every $x \in K^{\ast}$, i.e. such that every element of the multiplicative group is a root of $x^4 - 1$? Of course the finite fields of order $3$ and ...
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Show $a^2 + b^2 + 1 \equiv 0 \mod p$ always has a solution if $p = 4k+3$

If $p = 4k+3$ is a prime number (so $p = 7,11,19$ but not $p = 5,13$ or $p =15$) then there are numbers $a,b$ such that: $$a^2 + b^2 + 1 \equiv 0 \mod p$$ For example $2^2 + 3^2 + 1 = 14 = 7 \times ...
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Proof that $(u+v)^p = u^p + v^p$ in vector space over finite field of characteristic $p$.

Let $\mathbb F_q$ with $q = p^n$ a finite field of characteristic $p$. Then for all $x,y \in \mathbb F_q$ we have $(x+y)^p = x^p + y^p$. If $V$ is a finite-dimensional $\mathbb F_q$-vector space of ...
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How to determine if a number is quadratic residue over binary field?

I am trying to determine if $x$ is quadratic residue over a binary field ($GF(2^n)$). For finite field of type $GF(p)$ (where $p$ is prime), one can find the answer by calculating $x^{(p-1)/2}$ and ...
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$X^p - t$ irreducible in $K[X]$ with $K:=\text{Quot}(\mathbb{F}_p[t])$

I've heard a teacher say that this also follows by the Eisenstein criterion by checking with $t$. But then $t$ would have to be prime in order for Eisenstein to have any validity, but $t$ is ...
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Counting the solutions of a quadratic equation

I have read that a non-singular conic will contain $p+1$ points on the finite field $\mathbb{F}_{p}$, but there is a exercise on Silverman's Rational Points on Elliptic Curves, p.142, 4.8 that tells ...
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What is $\rm{Aut}(\rm{AGL}(1,q))$ for $q=p^r$ and $r > 1$?

Let's $K$ be a finite field with $q=p^r$ elements where $p$ prime is the characteristic of $K$. We denote $\rm{AGL}(n,q)$ the affine general linear group of dimension $n$ over $K$. I'm trying to ...
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56 views

Are two dot products of a random variable vector independent?

Let $w,v$ be two different vectors in the finite vector space $Z_p^m$ over $Z_p$ where $p$ is prime. Let $u$ be a vector chosen uniformly at random from $Z_p^m$. Are the random variables $u \cdot w$ ...
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56 views

Which one of these rings are isomorphic?

We have the follwoing ring and we need to see which one are isomorphic:- $\mathbb{Z[i]/(5)}$ $F_{5}[X]/(X^2-1)$ $F_{5}[X]/(X^2+1)$ First I thought the first one is field. but later based on answer ...
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2answers
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System of quadratic equations over field of size 2

I am working on system of quadratic equations. \begin{cases} (\alpha_1^1x_1+\ldots+\alpha_n^1x_n)(\beta_1^1y_1+\ldots+\beta_m^1y_m)=0\\ \ldots \\ ...
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37 views

When is a Hermitian matrix of the form $g^\circ g$ for some matrix $g$.

I'm trying to figure out some properties of Hermitian matrices over finite fields. Namely, let $K_0=\mathbb{F}_q$ be a field with $q$ elements, and let $K=\mathbb{F}_{q^2}$. The matrix algebra ...
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1answer
31 views

How to invert matrix in finite field

I want to invert matrix $A$ in the finite field $\mathbb{F} = \mathbb{F}_2[x]/p(x)\mathbb{F}_2$ with $p(x)=x^8+x^4+x^3+x+1$. This finite field is used by the encryption scheme AES. $A = ...
4
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3answers
65 views

Reducibility of polynomials modulo p [duplicate]

How do you show that $X^4-10X^2+1$ is reducible modulo every prime p? I've managed to show it for all primes less than 10, for primes greater than 10 we have $X^4+(p-10)X^2+1$. Where do I go from ...
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Division in finite fields

Let's take $GF(2^3)$ as and the irreducible polynomial $p(x) = x^3+x+1$ as an example. This is the multiplication table of the finite field I can easily do some multiplication such as ...
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Irreducible of $\mathbb Z/p\mathbb Z$.

What are the irreducible number of $\mathbb Z/p\mathbb Z$ ? It looks strange since in a field it looks complicate to talk about irreducible since all element are invertible. So if my question has no ...
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Compute Galois groupe of $\mathbb F_q/\mathbb F_p$

I have to compute Galois group of $\mathbb F_q/\mathbb F_p$ where $q=p^n$. I know already that $\mathbb F_q/\mathbb F_p$ is galois, so I don't need to prove it. Moreover, I know that ...
2
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36 views

Automorphisms of the affine semilinear group $A\Gamma L(1,2^{n})$

In this question, it is mentionned that the group of automorphisms of the semilinear group $A\Gamma L(1,2^{n})$ is the group itself. Do you have a short proof of this fact?
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29 views

The field $\mathbb F_q$ is it $\cong \mathbb F_p[X]/(X^q-X)$?

I know that $\mathbb F_q$ where $q=p^n$ is a field with $q$ element and that is by definition the splitting field of $X^q-X$ Q1) Is it really by definition ? Can we say that $\mathbb F_q\cong ...
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Question on proof about finite fields mentioned on wikipedia

In the first paragraph of wikipedia:Finite fields they write The identity $$ (x + y)^p = x^p + y^p $$ is true (for every $x$ and $y$) in a field of characteristic $p$. For every element ...
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1answer
20 views

For each $x \in GF(p^{2n})$ it is $1 - x^{p^n + 1} \in GF(p^n)$

Let $GF(p^{2n})$ be a finite field of order $p^{2n}$. Then $GF(p^n) \subseteq GF(p^{2n})$. Why do we have $1 - x^{p^n+1} \in GF(p^n)$? I know that $x^{p^n} - x = 0$ holds iff $x \in GF(p^n)$, but why ...
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32 views

Pth Root of Polynomial Over Finite Fields for Yun's Algorithm

While I was implementing Yun's algorithm in java, I could not figure out an algorithm to find the $p$th root of a polynomial in $\mathbf{F}_p$ where the polynomial is a perfect power of $p$. How would ...
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In finite field why $\overline c = -c \ne 0$ for $\overline c = c^{p^f}$.

Let $GF(p^{2f})$ be a finite field of order $p^{2f}$. Consider the map $\overline x := x^{p^f}$ for $x \in GF(p^{2f})$. Let $b \in GF(p^{2f}) - GF(p^f)$ and set $c := b - \overline b$. Why do we ...
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Relation between the sum of the values of a polynomial $f$ over a finite field, and the additive character sum with $f$ as the polynomial argument

Let $F$ be a finite field, let $f(T) \in F[T]$ and let $\psi$ be the canonical additive character of $F$. If $\sum_{x \in F}f(x) = 0$, what can we say of $\sum_{x \in F} \psi(f(x))$?
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2answers
54 views

Why crypto algorithms are primarily based on finite fields?

I want to learn why people use finite fields in cryptography? I mean there are other fields like number fields, function fields that are not finite. There are also some other topological fields, like ...
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11 views

Primitive polynomial over $GF(2^n)$ with $n>1$

I need at least two primitive polynomials over $GF(2^n)$ (with $n>1$). All the articles I've found only primitive polynomials over $GF(p)$, never over $GF(p^n)$. Does it exist some tables of such ...
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16 views

decoding a block code in non systematic form

I've a Generator matrix in Gf(2) in non systematic form( no seperate, data bits, and no separate parity). It is a full rank matrix. Now to find H[7x20] from this matrix I used the property ...
2
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1answer
90 views

find the structure of an elliptic curve over a finite field

For the elliptic curves E1,E2,E3, and E4 defined below, determine the structure of the groups Ek(F13) by using the information given below together with a minimal amount of extra (hand) ...