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 minimal polynomial of primitive element (field theory)

I am given a primitive element $\alpha$ in the Galoisfield $F_{2^6}$. The question is to find the mimimal polynomial of $\alpha^7$. How to I find this? My thoughts so far: $$ \alpha^7 \rightarrow ...
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1answer
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Group ring of a cyclic group over a finite field

Suppose $ p $ a prime integer and $ n $ a positive integer. Does anyone know off the top of their heads if the group ring $ \mathbb{F}_{p}[\mathbb{Z}/n] $ (perhaps regarding $ \mathbb{Z}/n $ as the $ ...
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Finding Irreducible Polynomials of Elements in Finite Quotient Polynomial Field over Finite Field [on hold]

The question is the following: Let $E = F_2[X]/< x^3+x+1 >$. For each $a \in E$ find $irr(a, F_2)$, its irreducible polynomial over $F_2$. Thanks for the help.
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1answer
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Bound on Lynden words made of $q$ letters

Let $N(q,n)=\frac{1}{n}\sum_{d|n}\mu(n/d)q^d$ for $q$ positive integer. Is it true that $N(q,n)<q^n/n$? This is true for $q$ prime which corresponds to the number of monic irreducible polynomials ...
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36 views

How to find inverse of generator of a finite field?

I need to find the inverse of generator of finite field $\mathbb{F}_{2^4}$ with irreducible polynomial , $f(x)=x^4+x+1$ i.e. if $g=0010$ is the generator of this field then how to find $g^{-1}$?
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1answer
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Roots of $X^{l-1}+1$ in a quadratic extension $F_q$, $q=l^2$

Consider a finite field $F_q$ where $q=l^2$ ($l$ can be of the form $p^m$). Does $F_q$ has a root of $X^{l-1}+1$? As $X^l+X = X(X^{l-1}+1)$ we can show that $X^{l-1}+1$ splits if it has a root. This ...
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2answers
38 views

Finite fields, characteristics and the Fundamental Homomorphism Theorem

I am trying to make sense of this proposition. I am fine with part (a), for part (b) however, can you explain what the computation proves? Can you not verify a homomorphism by checking the 3 ...
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16 views

Is collinearity sufficient for a map to be affine on a cartesian product of finite fields?

Let $V=\mathbb{Z}_p\times \mathbb{Z}_p$, where $p$ is prime. Let $\pi:V\rightarrow V$ be a map such that if two lines are parallel in $V$, then their images remain parallel. This is a property of ...
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Let $F =\mathbb Z_3$. Give an example of irreducible monic polynomial in F[x]. [on hold]

Let $F =\mathbb Z_3$. Give an example of irreducible monic polynomial in F[x]. Then give an extension field $\mathbb Z_3 < H$ and classify according to the fundamental theorem of abelian groups. ...
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3answers
51 views

Finite field definition question

My textbook says that $ (\mathbb{Z}_{m},+,*)$ is a field if and only if m is a prime number. However, on Wikipedia it says: "Finite fields only exist when the order (size) is a prime power $p^{k}$ ...
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Number of solutions of $x^2 + y^2 + z^2 = 0$ over finite fields.

I want to prove that the number of elements of the set $\{(x,y,z)\in \mathbb{F}_p^3: x^2 + y^2 + z^2 = 0\}$ is $p^2$. I know that the number of elements of the set is a multiple of $p$ using the ...
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1answer
99 views

Irreducible Polynomials over Finite Fields [on hold]

How would I show that $p(x)=x^5+x^2+1$ is an irreducible polynomial over $\Bbb Z_2=\{0,1\}$.
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1answer
19 views

Derivative of a polynomial in finite fields

In Introduction to finite fields and their applications by R.Lidl, the definition of the derivative for a polynomial such that ($a_i\in GF(q)$) $$f_{(x)}=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$ is ...
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23 views

Gram-Schmidt in characteristic two?

I was helping someone work on a computing problem with bit vectors that reduced to finding a basis knowing a spanning set, and realized quickly that the Gram-Schmidt process does not work as expected ...
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25 views

Why Fibonacci LFSR random number generation works?

If I use primitive polynomial of GF($2^m$) in Fibonacci LFSR, it is generating all m-length binary combinations. But, I cannot understand why this should happen. I am not getting any mathematical ...
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2answers
37 views

A subgroup of $\textrm{GL}(3,q)$ of order $q^2(q-1)$

Let $q$ be a prime power. Consider the multiplicative group $\textrm{GL}(3,q)$ of the $3 \times 3$ matrixes with coefficients in $\mathbb{F}_q$ which are invertible. The matrixes $$ M_{a,b,c} = \left( ...
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Can the integers be made into a vector space over any Finite Field?

Given a Finite Field $F$, can the the abelian group $\mathbb Z$ be made into a vector space over $F$ without changing the additive structure of $\mathbb Z$? This seems like it shouldn't be ...
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1answer
36 views

Methods for factorising polynomials into irreducibles over finite fields

I was given a problem recently, part of whose solution was to factorise $x^{15}+1$ in $\mathbb F_2[x]$. It turns out that the factorisation is $$(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1),$$ ...
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1answer
36 views

How can one solve an equation over over a specific finite field?

How can one solve an equation of the following form where the coefficients are in $GF(2^{128})$? $Az^3 + Bz^2 + Cz + D = 0$ The operations are defined over the same field.
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1answer
31 views

Minimal polynomials

Can someone explain to me how the minimal polynomials in page 4 of this document are obtained? Please help me. http://web.ntpu.edu.tw/~yshan/BCH_code.pdf It should be something standard about ...
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1answer
18 views

Proof of primitive roots in $F_{128}$

What would be the simplest way to prove that every element in $F_{128}$ is a primitive root except zero $(0)$ and the identity. Well, clearly 0 can not be a primitive root, and i also know that ...
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1answer
29 views

Finite fields and cardinality

I am trying to get my head around the proof of the following: Suppose K is a finite field. With $p=charK, |K|=p^r$ where r is a positive integer. I am supplied with the following proof: I do not ...
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The number of subspaces in a finite field

In my lecture note, there is the following statement: The number of 2-dimnesional subspaces in $F_2^3$ equal to the number of 1-dimnesional subspaces in $F_2^3$ Here, $F_2^3=F_2 \times F_2 \times ...
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1answer
38 views

Binomial of Mersenne prime power.

Let $f(x)$ be irreducible over $\mathbb{Z}_2$ of degree $p$, where $p$ is prime. Let $2^p-1$ be a Mersenne prime. I have to show \begin{equation*} f(x) \mid (x^{2^p-1}-1). \end{equation*} I am ...
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1answer
45 views

Elements of a finite field

What is the proof that for any given element $c$ of $F_q$, there exist two elements $a$ and $b$ of $F_q$ such that $a^2 + b^2 = c$. i know that $q$ is the characteristic of this field, but i don't see ...
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If $p$ is a prime, prove there are exactly $\frac{p^3-p}{3}$ monic irreducible cubic polynomials in $\mathbb{Z}_p[x]$

If $p$ is a prime, prove there are exactly $\frac{p^3-p}{3}$ monic irreducible cubic polynomials in $\mathbb{Z}_p[x]$ I am looking some notes here but don't know in general how to approach this ...
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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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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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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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45 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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63 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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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
39 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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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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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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1answer
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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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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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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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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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
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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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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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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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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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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 ...