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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How to find the power of generator defined over finite field , $\mathbb F_{2 ^m}$?

List item Actually ,I am trying to execute the algorithm to find the power of generator of field(group) as shown in table of attached file but when k=7 and onward ,I could not understand what is ...
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71 views

Sum of elements of a finite Field

Let $F$ be a finite field and $i$ an integer. Calculate the sum of all the elements of $F$,each raised to the $i-th$ power. My approach so ...
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4answers
537 views

Why do Z/7 have no cubic root of 2?

I was reading a textbook and came across the following line: Now we prove there is no cube root of 2 in $Z/7$. By noting that $(Z/7)^\times$ is cyclic of order 6, it will have only two third ...
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Efficient computation of matrix determinant in finite ring

I am trying to implement generalization of Hill cipher. My idea is very simple: the size of key matrix should be arbitrary number not only three. All steps of this cipher is trivial except computation ...
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2answers
51 views

Infinite non-abelian $ p $-groups.

Is it true that every nilpotent group is a solvable group? It is true for finite $ p $-groups, but I am not sure about infinite $ p $-groups.
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72 views

Squares in a Finite Field [duplicate]

Show that in any finite field,each of its elements can be written as the sum of two squares. Well,I hate to admit-this being also my first post-that I have not proven it yet.I tried to work on the ...
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49 views

Checking irreducibility of a polynomial over a finite field

A part of a coding theory course I am doing includes some questions on irreducible polynomials. I have a question with solution but am worried I have interpreted it incorrectly. So for $\mathbb F_5$ ...
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1answer
41 views

How does GAP understand $SL_2(\mathbb{F}_3)$?

When one is asking for " IrreducibleRepresentations(SL(2,3))" it makes sense that the matrices it returns are over the complex field as thats the field on which the representations are. There GAP ...
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37 views

Primitive element of finite field

I am looking for primitive element of galois field of order $8.$ So, I can look at the field $\mathbb{F}_8=\mathbb{Z}_2[x]/(x^3-x-1)$. I computed $\mathbb{F}_8^{\times}$ and now the primitive ...
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57 views

Composition of polynomials over finite fields

Consider the set of polynomials of degree at most $n$ over a finite field $k_q$ with $q$ elements where $q$ is prime: $$ P_{n,q} = \left\{ x + c_2 x^2 + \cdots + c_n x^n:\ c_i \in k_q \right\}. $$ It ...
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23 views

Reducible Polynomials in finite fields

I am stuck on the following question. Verify that $x^5 + x + 1$ is reducible in $Z_2[x]$ and find its factors. Help would be much appreciated whether it is the answers with how you did it or just the ...
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Proving that $SL_2(\mathbb{Z}_5) / \{\pm I\}\simeq A_5$

I see here that one can prove that $$ SL_2(\mathbb{Z}_5) / \{\pm I\} \simeq A_5 $$ using the First Isomorphism Theorem. My question is how one would do that. I know that I need a surjective ...
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2answers
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What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?

It seems that there is a action by which $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$ permute the $p^2$ ordered tuples in $\mathbb{F}_p^2$. What is the map from the $2 \times 2$ matrices over ...
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1answer
71 views

Find Primitive Root for Polynomial Field

Can someone help me get started on the problem below: Recall that $\mathbf{F}_{p^k}$ can be realized as $\mathbf{F}_p[x]/P(x) \cdot \mathbf{F}_p[x]$ where $P(x)$ is a polynomial of degree $k$ with ...
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81 views

How to read GAP's output on “IrreducibleRepresentations”?

For example for the group $SL_2(\mathbb{F}_3)$ I get the following, ...
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35 views

Find the # of elements of $F$ that can be written in the form $a^n-a$.

Im trying to show that for a finite field $F$ or order $ p^n $ there are $p^{n-1}$ elements that can be written in the form $a^p-a$ for some $a\in F$ I know that if we consider $x^p-x$, then ...
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Is there a ring - homomorphism $\mathbb{F}_p \rightarrow \mathbb{F}_q $ (p,q prime , $p \not= q$ )?

So we have two prime fields and seek a homomorphism between them. I assume that i have to find a homomorphism that is valid for all p,q prime , $p \not= q$, not just one for each choice. I would say ...
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3answers
53 views

Irreducibility of polynomials over finite field of integers $\bmod 11$.

Theorem (Fermat's little Theorem). If $p$ is a prime and $a \in \mathbb{Z}$ with $a \nmid p$ then $a^{p-1} \equiv 1 \mod p.$ Let $\mathbb{Z}/p\mathbb{Z}$ denote the multiplicative group of ...
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2answers
30 views

Example of formally real field $F$ with irreducible polynomial of odd degree in $F[x]$

Let $F$ be a formally real field; i.e. $-1$ cannot be expressed as the sum of squares. Let $K/F$ be a field extension of odd order. Hence, there exists $\alpha\in K$ which satisfies an irreducible ...
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36 views

Prove that $|V| = |F|^{\dim V}$ for a finite dimensional vector space $V$ over $F.$

In my algebra class, we proved that a quotient ring $F[x]/(f(x))$ is a vector space over $F$ and $\dim_F F[x]/(f(x)) = \deg f(x).$ I am attempting to use these facts to prove that the field $U = ...
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67 views

Irreducibility of $x^{2n}+x^n+1$

I want to know for what $n$, $$x^{2n}+x^n+1$$ is irreducible modulo 2. I think for $n=3^k$ but have no idea how to prove it.
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1answer
45 views

Showing explicitly that $\alpha \mapsto \alpha+1$ is an automorphism in the Galois group of $x^p-x+a$?

I have already shown that $x^p-x+a$ is irreducible and separable over $\mathbb{F}_p$ and I have shown that if $\alpha$ is a root, so is $\alpha+1$, so $\alpha$ generates the extension, i.e. the ...
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Finding roots in finite fields.

On pg. 587 (in the finite fields chapter) of Abstract Algebra, 3rd ed. by Dummit and Foote, the following statement is made: 'If $f_1(x)=x^4+x^3+1$, $f_2(x)=x^4+x+1$ are two of the irreducible ...
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33 views

Factor Ring question and finding maximal ideals of $\mathbb{Z}\times\mathbb{Z}$

What is the maximal ideal of $\mathbb{Z}\times\mathbb{Z}$? I think since $(\mathbb{Z}\times\mathbb{Z})/(\{0\}\times\mathbb{Z})$ is isomorphic to $\mathbb{Z}$, it seems like that ...
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1answer
23 views

Order of a polynomial in $\mathbb F_q[x]$

I came across the term "order" in the context of $\mathbb F_q[x]$, specifically of irreducible polynomials. Does this mean order in the group theoretical sense? I tried to prove that every polynomial ...
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1answer
54 views

Proving the number of solutions in a finite field

Let F be a finite field containing $q$ elements. Let $a \in F$ be nonzero. If n divides $q -1$ prove that $x^n - a = 0$ has either n solutions in F or no solutions in F. I am unsure of how to begin ...
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1answer
34 views

Quotient by power of maximal ideal

Suppose $R$ is a commutative ring (but see the edit portion below) and $\mathfrak{m}$ is a maximal ideal of $R$ such that $|R/\mathfrak{m}|<\infty$. Also assume that $k$ a positive integer. Is ...
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32 views

Finite field, how to satisfy equation?

Suppose we have $\mathbb{F}_{2^5}$ defined by polynomial $x^5+x^2+1$, and (this is homework exercise, which I kinda solved) it is required to find suitable elements $b$, so that it satisfies equation ...
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37 views

How to show mutually orthogonal latin squares

I have a question concerning mutually orthogonal latin squares (MOLS). Let $ \mathbb F $ be a field of $n\in\mathbb N$ elements. For all $q\in\mathbb F \backslash \{0\}$, define $n\times n $ tables ...
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Solution of system of equations in prime fields

In 'Algebra', Artin writes that the system of equation: $$8x+3y = 3$$ $$2x+6y = -1$$ have no solutions in $\mathbb{F}_2$ and $\mathbb{F}_3$ as the determinant (of the coefficient matrix) evaluates ...
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37 views

$x^{p^{k}-1}-1$ divides $x^{p^{n}-1}-1$ in $\mathbb{F}_{p}$ iff $k$ divides $n$

Let's consider two polynomials in $\mathbb{F}_{p}[x]$: $f(x)=x^{p^{n}-1}-1$ and $g(x)=x^{p^{k}-1}-1$. How to prove that $g(x) \mid f(x)$ iff $k \mid n$? For instance, it's worth trying this: ...
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How to prove this is a field?

Let $F=(\mathbb{Z}/5\mathbb{Z})[x]/(x^2+2x+3)$. How do I prove $F$ is a field? I've shown its a commutative ring with an identity $\bar1$. Then we let ...
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38 views

How do I prove that $X^{p^n}-X$ is the product of all monic irreducible polynomials of degree dividing $n$?

How do I prove that $X^{p^n}-X$ is the product of all monic irreducible polynomials in $\mathbb Z_p[X]$ of degree dividing $n$? Let $\bar Z_p$ be an algebraic closure of $Z_p$. Define $F=\{x\in ...
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35 views

Finite fields and isomorphism

For each prime number p, let $F_p$ denote the field of integers modulo p. Now let K be any finite field. a) Prove that K contains a subfield isomorphic to $F_p$ for some prime number p b) Prove that ...
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On the existence of a polynomial in several variables with only one zero

Given an ordered field $\mathbb{K}$, then for all $n>1$ there exists $f\in\mathbb{K}[X_1,\ldots,X_n]\ s.t.\ \mathcal{V}^{\mathbb{K}^n}(f):=\{p\in\mathbb{K}^n:f(p)=0\}=\{(0,\ldots,0)\}$ For ...
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Proving the “freshman's dream”

$(x+y)^p = x^p + y^p$ holds in any field of characteristic $p$. However all the proofs I have seen use induction and some relatively nasty algebra despite how fundamental this fact seems. What is the ...
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39 views

Given $f=x^4+x+1 \in \mathbb Z_{2}[x]$ is primitive, write down an $m$-sequence ${a_n}$ associated to $f$

I'm not sure how to solve this question exactly. I know that the period will be 15 but I don't know how to construct the $m$-sequence.
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1answer
42 views

T is multliplication by a primitive element

Consider $\Bbb F_{p^n}$ as a vector space over $\Bbb F_{p}$. Now let $T$ be a non-zero $\Bbb F_{p}$-linear map from $\Bbb F_{p^n}$ to $\Bbb F_{p^n}$. Now if $0$ and $\Bbb F_{p^n}$ are the only ...
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1answer
39 views

About ring $( \frac{F[t]}{t^2})$

I was trying to study some properties of the group $GL_2 ( \frac{F[t]}{t^2})$ where $F$ is a field with $p$ elements. $p$ is a prime. I found that $ \frac{F[t]}{t^2}$ is a discrete Valuation ring. ...
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82 views

The Galois closure of $\mathbb{F}_2(x,y) / \mathbb{F}_2(x)$

Consider the field extension $\mathbb{F}_2(x,y) / \mathbb{F}_2(x)$ where $y$ is a root of the polynomial $g(T) = f(x,T) \in \mathbb{F}_2(x)[T]$, with $$ \begin{array}{l} f(x,y) = x^{12} + x^{10}y^2 ...
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about hyperplanes in finite fields

I was reading an article that said: if I have a finite field, say $\mathbb{F}_q^k$ where $q=p^n$ and $p$ is a prime; and a (k-2)-dimensional subspace, say $U\subset \mathbb{F}_q^k$ given by the span ...
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69 views

Rijndael S-Box algorithm: Can someone please explain how this code calculates the multiplicative inverse?

Wikipedia's explanation of the Rijndael Cipher's S-Box gives c code for calculating the S-Box. I've been able to calculate the S-Box values using exponent and log look-up tables to calculate the ...
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1answer
31 views

Isomorphisms between finite fields of same characteristic [duplicate]

Here I am taking the definition of isomorphism to be an injective homomorphism. Suppose we have two finite fields of the same characteristic, $\mathbb{F}_{p^n}$ and $\mathbb{F}_{p^m}$ with $m<n$. ...
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1answer
192 views

Reduction modulo p of a linear group over the rational numbers

A paper (http://arxiv.org/pdf/1407.3158v2.pdf) contains the following theorem: Suppose $\mathbb{G}$ is a connected, simply connected, semisimple algebraic group defined over $\mathbb{Q}$, and let ...
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27 views

“Consecutive” square residues in odd-order finite fields

Let $\mathbb F = \mathrm{GF}(p^r)$ be finite field for $p$ an odd prime, and define $$ \begin{align} Q &= \bigl\{ u^2 \,\big|\, u \in \mathbb F^\times \bigr\} \\ N &= \mathbb F^\times ...
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69 views

$\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ is a Galois extension, with $\mathbb{F}_4(x,y)$ a function field of an algebraic curve

Consider the field extension $\mathbb{F}_4(x,y)/\mathbb{F}_4(x)$ where $y$ is a root of the polynomial $$ f(T)= x^4 + x^2T^2 + x^2T + x^2 + xT^2 + xT + T^4 + T^2 + 1 \in \mathbb{F}_4(x)[T]. $$ ...
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143 views

Differentiate polynomials in $\mathbb{Z}_2[x]$

It seems suggested that the differential of a polynomial in $\mathbb{Z}_2$ is as I would expect: $$\begin{align} &f = x^6 + x^3 + x + 1 \\ &f' = 6x^5 + 3x^2 +1 \mod 2 \\ &f'= x^2 + 1 \\ ...
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Subset of numbers analogous to primitive polynomials over finite fields

It is well known that many problems in number theory have an analogue on the ring of polynomials over finite fields and vice versa, the primes in $\mathbb{F}_q[x]$ being the irreducible polynomials. ...
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1answer
24 views

Prove that $F[x]/(f(x))$ has $q^n$ elements [duplicate]

let $F$ be a finite field of order $q$ and let $f(x)$ be a polynomial in $F[x]$ of degree $n \geq 1$. Prove that $F[x]/(f(x))$ has $q^n$ elements. I know that if a field is finite then the order is ...
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1answer
64 views

About Cayley graphs on finite fields.

If one is given $n$ vectors of length $n$ $\in \mathbb{F}_{p^k}^n$ for some prime number $p$ and $k \in \mathbb{Z}^+$ then how can one check if they are linearly independent? (the issue is if there ...