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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Prove identity in quotient group

I'm studying for my algebra exam, and came across the following problem, which I'm not sure how to solve Let $f = X^2 - 1 \in \mathbb{F}_3[X]$ and $\alpha = X + \langle f \rangle \in ...
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133 views

Proving $f = x^8+x^7+x^3+x+1$ to be irreducible over $\mathbf{F}_2$.

I am taking all the irreducible polynomials over $\mathbf{F}_2$ of degree 1 to 4 inclusive and using the long division to determine whether the given $f$ is divisible by anyone of them and it is not ...
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27 views

Finding the order of an irreducible polynomial $f$ in $F_3[x]$ of degree 4?

The technique I am using is based on the long division of $x^e - 1$ (e is to be the order) which is really tiresome. So what the other methods (efficient)?
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44 views

Quotient group element is a unit

I'm studying up for my algebra exam, and I'm not exactly sure how to solve a problem like the following Let $f = X^2 + 1 \in \mathbb{F}_5[X]$, $R = \mathbb{F}_5[X]/\langle f \rangle$ and $\alpha = ...
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83 views

Galois group of $X^4+X^3+1$ over $\mathbb{F}_4$

I'm confused. Realizing $\mathbb{F}_4=\mathbb{F}_2[T]/(T^2+T+1)$, the polynomial $X^4+X^3+1$ splits as $(X^2+TX+T)(X^2+(T+1)X+T+1)$. These 2 factors have no root over $\mathbb{F}_4$, so they're ...
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irreducibility of $x^{5}-2$ over $\mathbb{F}_{11}$.

I am tasked to show that $x^{5}-2$ is irreducible over $\mathbb{F}_{11}$ the finite field of 11 elements. I've deduced that it has no linear factors by Fermat's little theorem. But showing it has no ...
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188 views

automorphisms of a finite field

Let $F$ be a finite field of characteristic $p$ over $\mathbb{Z}_p$ with $p^r$ elements. Then I have proved that the map $\phi: x\mapsto x^p$ is a field automorphism of $F$. Moreover, $\phi(x)=x$ if ...
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40 views

most efficient way to find distinct complimenting subspaces over a finite field

Let $V$ be a $n$-dimensional vector spaoce over $\mathbb{F}_p$ and let $W$ be a $k$-dimensional subspace. What's the most efficient way to algorithmically write down a basis for each distinct ...
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54 views

classification of $2$-dimensional field extensions

Let $F$ be a field and $K:F$ be a field extension such that $[K:F]=2$. Then (i). If $Char(F)\neq 2$, then there exists $\alpha\in K^*$, $\alpha\notin F^*$, such that $K=F(\alpha)$ and $\alpha^2\in ...
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example of Frobenius endomorphisms that is not automorphism

Let $F$ be a field of characteristic $p$ and $F$ has infinitely many elements. Prove that the map $\sigma: \alpha\mapsto \alpha^p$ is a field endomorphism but not necessarily automorphism. How to ...
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51 views

Prove that $K\times K[X]/(X^7-1)\cong K\times \dots \times K$

Given that $K$ is a finite field of order $q\equiv1\text{ mod } 7$, I have to prove that $$K\times K[X]/(X^7-1)\cong K\times \dots \times K\ (8 \text{ times } K).$$ It's the same to prove that ...
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1answer
93 views

If $F$ is a finite field then $|F|=p^n$ for prime $p$ and some integer $n$ [duplicate]

I need some help proving the following: If $F$ is a finite field then $|F|=p^n$ for prime $p$ and some integer $n$. By contradiction, suppose $|F| =pq$ for primes $p$ and $q$. Then by Cauchy's ...
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55 views

Splitting field over $\mathbb{F}_3$

The splitting field of $f(x)=x^8-1$ over $\mathbb{F}_3$ is $\mathbb{F}_{3^d}$ where $d=ord_{(\mathbb{Z}/8\mathbb{Z})^*}(3)=2$. But $f(x)=(x^4+1)(x^4-1)$ and $x^4+1$ is irreducible over ...
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43 views

If $\alpha$ = $\beta^q - \beta$ where both $\alpha$ , $\beta $ belongs to $F_q^n$ which is extension of $F_q$

Clearly $\beta$ is a root of $f(x) = x^q - x - \alpha$ and the other roots are its conjugates w.r.t $F_q$ so $f(x)$ splits in $F_q^n$ . But the degree is q so there are q distinct roots and my problem ...
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32 views

Two ways of computing trace? (correct/incorrect)

Suppose f(x) is an irreducible polynomial of degree 2 over $F_3$ and $\alpha$ is a root in $F_9$ and we have to compute the trace of $\alpha^4$. One way is to compute by the definition (sum of the ...
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70 views

Count the number of rational canonical form&find similarity classess

For a finite field $F=F_q$ having $q=p^d$ elements ($p$ a prime integer), compute the number of similarity classes in the vector space $M_n(F)$ of $n\times n$ matrices over $F$. (Maybe count the ...
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40 views

Why is it true that $F_{q^n} = F_q(\alpha)$ where $\alpha$ is the primitive element of $F_{q^n}$?

Since $\alpha$ is the primitive element of $F_{q^n}$ then $F_{q^n} = \{0, \alpha, \alpha^2,\cdots, \alpha^{q^{n -2}} , 1\}$. Then how $F_q(\alpha)$ is equivalent to $F_{q^n}$? Because what I ...
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How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
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92 views

Ultraproduct of the algebraic closure of finite fields

Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, ...
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51 views

Show a polynomial is irreducible

I'm working through the proof of Hasse's theorem and I think I need to show that the polynomial $x^4 - 2ax^2 - 8bx + a^2$ is irreducible over $\mathbb{F}_p$, where $a$, $b$ are integers and $p$ is ...
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52 views

Structure of $C(F_5)$ from Rational Points on Elliptic Curves

In the book Rational Points on Elliptic Curves by Silverman/Tate one examines the elliptic curve $y^2 = x^3 + x + 1$ over $F_5$. One can then easily determine the group $$ C(F_5) = \lbrace ...
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66 views

polynomial over a finite field

Show that in a finite field $F$ there exists $p(x)\in F[X]$ s.t $p(f)\neq 0\;\;\forall f\in F$ Any ideas how to prove it?
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52 views

Find a finite extension of $\mathbb F_2(x,y)$ having infinitely many intermediate subfields.

Let $\mathbb F_2$ be a field of order $2$. Let $F$ be the function field $\mathbb F_2(x,y)$, i.e., $x,y$ are independent variables and $F$ is the field of quotients of the polynomial rings $\mathbb ...
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2answers
49 views

Linear independence over $\mathbb{F}_p$ for varying primes

Let $v_1,\ldots,v_k\in \mathbb{F}_p^n$ be a set of vectors, where $p$ is a prime. Assume further that the components of each vector can be represented by integers smaller than some integer $k$. Is ...
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44 views

What's the connection between irreducible polynomials and fixed-frobenius elements in a finite ring?

Consider the ring $A_{p,n} = \mathbb{F}_p [x]/ (x^{p^n}-x)$. It has a basis $\{1, x, x^2, \ldots, x^{p^n - 1}\}$. The Frobenius endomorphism $x \mapsto x^p$ permutes elements of this basis. I've ...
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Finding the smallest $k$ such that $f(x)$ divides $1-x^k$ where $f(x)$ is over $\mbox{GF}(2)$?

One technique is iterative that is to assume alpha as the root and solve for a higher exponent ($x$) until $\alpha^{x} = 1$. Is there any other technique?
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54 views

Finding a primitive element of a finite field

Let $F = \mathbb{F}_p[x]/(m(x))$, where $m(x)$ is irreducible in $\mathbb{F}_p[x]$. How do I find a primitive element of $F$, i.e., one that generates the nonzero elements of $F$ multiplicatively? ...
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57 views

Multiplication in $\text{GF}(2^n)$

Computing products in $\text{GF}(2^n)$ involves choosing a prime modulus polynomial to "reduce" by. In the case that there are several such polynomials to choose from, do they all yield identical ...
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60 views

Units in finite polynomial rings

Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where ...
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Units in finite polynomial rings [duplicate]

Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where ...
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52 views

$F$ a finite field of $p^n$ elements. Suppose $F^\times=<x>$. Then $\phi(x)=x^{p^r}$ for automorphisms

Let $p$ be a prime and $F$ a finite field of $p^n$ elements. Suppose $F^\times=<x>$. Let $\phi$ be an automorhpism of $F$. Then prove that $\phi(x)=x^{p^r}$ for some integer $r$. How to prove? ...
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27 views

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$.

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$. It suffices to prove for the case $F=\mathbb{Z}_p$. How to prove?
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143 views

How to find null space of a matrix over $\mathbb{GF}(2)$?

How to find null space of a matrix over $\mathbb{GF}(2)$ ? Are there any algorithms available? I am not sure here is appropriate to ask this or not: Is there any routines for MATLAB or Maple?
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109 views

Discrete logarithm tables for the fields $\Bbb{F}_8$ and $\Bbb{F}_{16}$.

The smallest non-trivial finite field of characteristic two is $$ \Bbb{F}_4=\{0,1,\beta,\beta+1=\beta^2\}, $$ where $\beta$ and $\beta+1$ are primitive cubic roots of unity, and zeros of the ...
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33 views

$K\supset \mathbb{F}_q$, $h=x^q-x+a$, $a\in K$ if $K$ is finite $h$ is reducible

$q=p^n$, $K\supset \mathbb{F}_q$, $h=x^q-x+a\in K[x]$ if $K$ is finite $h$ is reducible. Let $L$ the splitting field of $h$ over $K$. Attempt: I proved that if $\beta$ is a root of $h$ and $h$ ...
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45 views

Is this a generator of a cyclic group?

Let $F=\mathbb Z/(p)$, where $p$ a prime number, $f(x)$ a monic irreducible polynomial in $K=F[x]$ of degree $n$, $K=F[x]/(f(x))$, and $E$ the multiplicative group of nonzero elements of $K$. Then ...
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143 views

Irreducible polynomial f in polynomial ring of finite field order n divides a particular polynomial if and only if degree of f divides n

Here is the problem I am attempting: Suppose that $K$ is a finite field with $|K| = q$. Show that if $f(x) \in K[x]$ is irreducible, then $f$ divides $g(x) = x^{q^n}-x$ in $K[x]$ if and only if ...
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Two questions on finite fields

I'm having some difficult with finite fields. If someone could point out a direction in which to look for these, or link to relevant material online, I would really appreciate it! I'm asked to factor ...
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multiplicative order in field

Let $\alpha$ be primitive element of GF(7). Then order of $\alpha$ is 6, i.e. $o(\alpha)=6$. Now we know that $\alpha^4$ is not equal to 1, and that $o(\alpha^4)$ = $\frac{6}{gcd(6,4)}$. This also ...
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The automorphism group of a field with $p^2$ elements

Suppose $K$ is a field. Then we call $f: K\to K$ a (field) automorphism if $f$ is a one-to-one, onto and unital (i.e. $f(1)=1$) homomorphism of rings. The following results are well-known. There ...
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Irreducible polynomials of degree $d$ such that $[x]$ generates $\mathbf{F}_{p^d}^{\times}$

It is often convenient to represent the field $\mathbf{F}_{p^d}$ as $\mathbf{F}_p[x]/(f(x))$, where $f$ is irreducible with degree $d$, $f$ has just a few nonzero terms, and $[x]$ itself is the ...
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Compute $a(x)b(x)+c(x)$ in $\mathrm{GF}(2^4)$ where the irreducible generator polynomial $x^4+x+1$.

Let the coefficients of $$a(x) = x^3+x^2+1,$$ $$b(x) = x^2+x+1,$$ $$c(x) = x^3+x^2+x+1$$ be in $\mathrm{GF}(2)$. Compute $a(x)b(x)+c(x)$ in $\mathrm{GF}(2^4)$ where the irreducible ...
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36 views

What is the smallest Finite Field in which the following polynomial is factorizable to irreducible factors?

What is the smallest Finite Field in which the following polynomial is factorizable to irreducible factors? $$(x^2+x+1)(x^5+x^4+1)(x^7+x^6+x^3+1) $$
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67 views

polynomials factorization over rings and finite fields

Any nonzero polynomial over a subring $R$ of $\mathbb{C}$ is a product of irreducible polynomials over $R$. And for any subfield $K$ of $\mathbb{C}$, factorization of polynomials over $K$ into ...
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133 views

Galois group of algebraic closure of a finite field

Is there any element of finite order of the Galois group of algebraic closure of a finite field, and if there is how can I construct it ? Thanks.
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Finite Fields: Linear Feedback Shift Register Algorithm Help

I am currently trying to generate a four digit Linear Feedback Shift Register with digits in Mod 5 using polynomials and finite fields. I am attempting to do so with the following algorithm: 1) ...
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1answer
58 views

Addition in finite fields

For a question, I must write an explicit multiplication and addition chart for a finite field of order 8. I understand that I construct the field by taking an irreducible polynomial in $F_2[x]$ of ...
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2answers
51 views

The equation $y^p-y = x$ in finite fields.

Define the trace by $$Tr \quad : \quad \mathbb{F}_{p^n} \ \longrightarrow \mathbb{F}_{p^n} \quad : \quad x \ \longmapsto \ x+x^p+\cdots +x^{p^{n-1}}$$ Now define yet another mapping: $$ L \quad : ...
3
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1answer
70 views

The trace map in a finite field.

Let $p$ be a prime number, and consider the mapping called the trace $$ Tr \quad : \quad \mathbb{F}_{p^n} \ \longrightarrow \ \mathbb{F}_{p^n} \quad : \quad x \ \longmapsto \ x + x^p + x^{p^2} + ...
2
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80 views

Number of solutions of $x^m \neq y^m$, $z^n=w^n=t^n$ over a finite field.

I am trying to compute the number of solutions of the following system of equations over a finite field $\mathbb{F}_q$ ($q$ may be considered odd prime power or just odd prime if needed): $$ x^m \neq ...