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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Suppose $F$ is a finite field of characteristic $p$. Prove $\exists u \in F$ such that $F = \mathbb{F}_{p}(u)$.

Suppose $F$ is a finite field of characteristic $p$ ($p$ a prime). Prove $\exists u \in F$ such that $F = \mathbb{F}_{p}(u)$. Here, $\mathbb{F}_{p}$ denotes the field with $p$ elements. Here is ...
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3answers
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Double finite field extension

Suppose we are given the field $\mathbb{F}_5$ and $p(X) = X^2-2 \in \mathbb{F}_5[X]$, an irreducible polynomial over $\mathbb{F}_5$. Let $\mathbb{K}$ denote the extension of $\mathbb{F}_5$ in which ...
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an irreducible polynomial over GF(2) is primitive over GF(2)

let $P \in F_{2} [X]$ of degree $7$, how to prove this: P is irreducible $\Leftrightarrow$ P is primitive i tried to use the mersenne prime !
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1answer
38 views

Compute the degree of the splitting field

I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial ...
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24 views

Describe all subgroups of $\operatorname{Inn}(\operatorname{GL}_2(\mathbb F_{p^n}))$

Is it possible to give a general discription of all subgroups of the group $\operatorname{Inn}(\operatorname{GL}_2(\mathbb F_{p^n}))$ of inner automorphisms of $\operatorname{GL}_2(\mathbb F_{p^n})$? ...
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1answer
19 views

When Errors Go Undetected in CRC

I understand that CRC will not be able to detect errors if: The remainder of $E(x) / G(x) = 0$ $E(x) = G(x).Z(x)$ for some polynomial $Z(x)$ I understand the first point, which means that if the ...
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38 views

Regarding Linear Subspaces over a Finite Field… TFAE:

Let $V=\mathbb{F}^n$, for a finite field $\mathbb{F}$. Prove the equivalence of the following statements: There is a linear subspace $C$ of $V$ with the property that every vector $v$ of ...
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15 views

Why does this criterion imply that $A$ is a subfield of $E$?

$E$ is an extension field of a field $F$ and $A$ is the subset of $E$ containing all the members algebraic over $F$. "To prove that $A$ is a subfield of $E$ it is enough to show that any two elements ...
2
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1answer
27 views

Elements of $\operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^k$

Let $p > 2$ be a prime number and $n\ge 1$ an integer, and consider the group $G = \operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^n(p^{2n} - 1)$. Let us denote by $\operatorname{Inn}(G)$ (the ...
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9 views

Characterise all subgroups of $\operatorname{SL}_2(\mathbb F_{p^n})$

The title pretty much explains everything. Is it possible to give an easy characterisation of all subgroups of $\operatorname{SL}_2(\mathbb F_{p^n})$?
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26 views

Number of orbits of the Frobenius automorphism

Let $q=p^s$ be a prime power congruent to $1$ modulo $4$, let $\mathbb{F}_q$ be the finite field with $q$ elements, and let $\phi$ denote the Frobenius automorphism (that is $\phi(a)=a^p$ for every ...
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2answers
35 views

Subgroups of $\mathbb F_{p^n}$

Is it possible to give a discription of the possible subgroups (with respect to $+$) of the finite field $\mathbb F_{p^n}$ (obviously, $p$ is a prime number). Of course, if $n = 1$, $(\mathbb F_p,+)$ ...
3
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1answer
36 views

Calculating zeta functions over a field

I am learning about zeta functions and have been trying the following example: Calculate the zata function of $x_0x_1-x_2x_3$ over $\mathbb{F}_p$. Does there exist an easy formula for calculating ...
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1answer
20 views

Order of an element

$\mathbb{F} = \mathbb{Z_2[x]/(x^3+x+1)}$ is a field. I need to find an element $a \in \mathbb{F}$ of order $p^n-1$ I know that $\mathbb{F}$ has order $2^3 = 8$ so $a$ must have order 7 ie, $a^7 = ...
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0answers
35 views

Solving system of equation over distinct finite fields

Is it possible to solve an equation systems which involves elements from distinct finite fields? One example could be elements from $\mathrm{GF}(2)$ and $\mathrm{GF}(2^2)$: With: $x=[1,1,0,1]$ ...
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1answer
37 views

polynomial over finite field, roots forming additive subgroup

Let $q=2^h$ and $t=2^r$ for some $h\ge r$ and denote by $\mathbb{F}_q$ the finite field of order $q$. (since the previous, simple version was wrong, I'm posting here a new version) Let $f$ be a ...
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3answers
47 views

Question about the Characteristic of $\mathbb{F}_{p^n}$

We can prove that any finite field of prime characteristic $p$ must have $p^n$ elements. Conversely, let $F$ be a finite field with $p^n$ elements, where $p$ is a prime number. Is the following ...
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1answer
32 views

Question about the cycle type of the Frobenius Automorphism

Let $f$ be an irreducible and separable polynomial of degree $n$ over $\mathbb{F}_p$. For a finite field $F$ of characteristic $p$, we define $\phi_p:\alpha\mapsto\alpha^p$. Then we know $\phi_p$ is ...
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2answers
44 views

How to prove a finite field is not ordered?

I have a set S={0,1}, and the addition and multiplication rules are \begin{array}{c|cc} +&0&1\\ \hline 0&0&1\\ 1&1&0 \end{array} \begin{array}{c|cc} *&0&1\\ \hline ...
4
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113 views

Polynomials over $\mathbb{F}_2$ with certain values in $\mathbb{F}_4$

Let $\mathbb{F}_4=\{0,1,u,u^2\}$ be the field with $4$ elements. Is there a polynomial $p \in \mathbb{F}_2[x,y]$ with the following property? (1) For $r,s \in \mathbb{F}_4$, we have $p(r,s)=u ...
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4answers
32 views

Calculating $3/10$ in $\mathbb{Z}_{13}$

I'm trying to calculate $\frac{3}{10}$,working in $\mathbb{Z}_{13}$. Is this the correct approach? Let $x=\frac{3}{10} \iff 10x \equiv 3 \bmod 13 \iff 10x-3=13k \iff 10x=13k+3$ for some $k \in ...
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31 views

Using Kronecker's theorem to construct a field with four elements

Use Kronecker's theorem to construct a field with four elements by adjoining a suitable root of $x^4-x$ to $\mathbb Z/2\mathbb Z$. Definition: A polynomial $f(x)\in F[x]$ splits over $F$ if it is ...
3
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1answer
43 views

Has anyone defined a limit of a sequence of fields? In particular, what is the limit of finite fields?

I'm curious about $$ \lim_{n \rightarrow \infty} \mathbb{F}_n $$ Is it $\mathbb{Z}$? That seems reasonable if you consider it as a set but of course $\mathbb{Z}$ is not a field so that is confusing. ...
4
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1answer
44 views

Exhibiting an isomorphism between two finite fields

So I want to find the isomorphism $\phi$ that takes $F = \mathbb{Z}_3/\langle x^3 - x - 1\rangle$ to $E = \mathbb{Z}_3/\langle x^3 - x + 1\rangle$. I understand that these are both finite fields of ...
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2answers
14 views

Implications of zero elemntary symmetric polynomials over a finite field

For a prime $q$ and an integer $n<q$, consider working over the finite field of $q^n$ elements. Denote by $s_n^k$ the $k$-th elementary symmetric polynomial in $n$ variables. That is, ...
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1answer
52 views

How to check field axioms given addition and multiplication tables

I need help with this question, i want to know the exact method of doing it with explanation. i am not able to get around with the logic of it.
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2answers
20 views

Homomorphisms between fields are injective.

How would I prove this? I know that I must show f(a)=f(b) => a = b I also know I must use the definition of homomorphism, ie: $f(a+b)=f(a)+f(b)$ $f(ab)=f(a)f(b)$ $f(1)=1$ I am assuming that a ...
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Automorphism of algebraic closure $\overline{{\bf F}}_p$.

Problem : I want to give an concrete example of automorphism of $\overline{{\bf F}}_p$ which fixes ${\bf F}_p$, where $$\overline{{\bf F}}_p =\bigcup_{n\geq 1} {\bf F}_{p^n} $$ and ${\bf F}_{p^n} $ is ...
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On the maximum number of polynomials in a certain subspace

I've already asked this question on mathoverflow, but no one answered. So I put this problem also here. Sorry for that. Let $\mathbb F_q$ be a finite field and let $e, k$ be positive integers with ...
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2answers
32 views

Solving quadratic equations in the field $F_5$

Let $y = x^2 + 2x + 2 = 0$. Solve the equation in the field $F_5$. So I used the common $b^2 - 4ac$ formula and got that $x$ is either $-1/2$ or $-3/2$ but I'm not sure if this is in the field...
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$+$ and $\times $ operations in finite fields are $+$ and $\times $ $mod$ some number

I don't know how to prove this: Addition and multiplication operations in finite fields addition and multiplication $mod$ some number. Also I have another doubt that as we know that a field acts ...
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0answers
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Roots of irreducible polynomials in a finite field

If $f$ and $g$ are irreducible polynomials over a finite field $\mathbb F_q$, both of degree $d$, then they both split in $\mathbb F_{q^d}$. One way to represent $\mathbb F_{q^d}$ is to adjoin a root ...
0
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1answer
28 views

If $K$ finite field of order $p^8$ where $p\ne3$ then $\sum_{\alpha \in K}{\alpha^2} = 0$

Let $K$ be finite field of order $p^8$ where $p\ne3$ is a prime. Show that $\sum_{\alpha \in K}{\alpha^2} = 0$.
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Square root for Galois fields $GF(2^m)$

Can we define a function similar to square root for $G = GF(2^m)$ (Galois field with $2^m$ elements) as $\sqrt{x} = y$ if $y^2 = y \cdot y = x$ ? For which elements $x \in G : \exists y \in G : y^2 = ...
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1answer
46 views

Field extension of degree 3 and polynomial roots

Deleted the old question, because tho whole question kind of changed. I am facing following problem: Given extension of finite fields $L/K$ of degree $3$, prove that every polynomial of degree $3$ ...
0
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1answer
25 views

number of quadratic residues in finite field

Is there a way to determine how many quadratic residues are there in the finite field $F_q$ for $q = p^k$? It seems if $q=p$ exactly $(p-1)/2$ are residues and the same amount are not. Does analogy ...
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1answer
45 views

Irreducible polynomial with LI roots

I am doing a small project under the domain cryptography. Recently I stuck with a problem in mathematics related to finite fields. My question is how can I found out irreducible polynomial(or ...
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2answers
62 views

Is the map an automorphism?

Please verify the following proof or comment on how would you have proven it. Suppose $q = p^2$ and we have $f: \mathbb{F}_{q} \to \mathbb{F}_{q}$ where $f(a) = a\cdot a^p$ Let $a\cdot a^p = b\cdot ...
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1answer
31 views

Constructing common extension of two finite fields

I am facing following problem, and would be very thankful for any input: I am supposed to prove that two finite fields with same number of elements are isomorphic. I know the usual proof via ...
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60 views

Jordan form of a Matrix with Ones over a Finite Field

Question: Find the Jordan Form of $n\times n$ matrix whose elements are all one, over the field $\Bbb Z_p$. I have found out that this matrix has a characteristic polynomial $x^{(n-1)}(x-n)$ and ...
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3answers
58 views

Can there be a Finite Field That Does Use Not Modular Arithmetic?

This may be a rather silly question, but I'm puzzled that (at least so far as I can tell) all finite fields use modular arithmetic. Is there no other way to construct a finite field than by defining ...
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0answers
32 views

Rational equality modulo $p$

Assume that we have two rational expressions $f,g\in \mathbb{Z}(x,y_1,\ldots,y_n)$ with the property that the variable $x$ can only appear in the numerator of $f$ while only in the denominator of $g$. ...
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How to show that the normalizer N of a Sylow $p$-subgroup P of $S_p$ in $S_p$ is the affine linear group AGL(1,$p$)?

I want to prove this as a special case of another fact that the normalizer of G in the group of set-bijections of G is isomorphic to the semi-direct product of Aut(G) and G itself. How do I link up ...
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1answer
60 views

Operations in $\mathbb{F}_{32}$

I've some difficulties about sums in the field $\mathbb{F}_{32}$. In particular I'm studying an example of a cryptographic attack, where there are a lot of sums in this field, which I don't ...
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1answer
15 views

Is a Spread Unique?

Let $V$ be a vector space of dimension $n$. It is well known that when $r | n$, there is a set of disjoint $r$-dimensional subspaces of $V$, which covers $V$, called Spread. My question is that is a ...
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Efficiently factoring polynomials over $\Bbb F_2$

I am attempting to write some software which is intended to generically answer the question of which Cyclic Redundancy Code (CRC) generating polynomial is used for a given set of sample messages using ...
3
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1answer
45 views

Square-free factorization of polynomials over finite fields

For any $f\in\mathbb{F}_q[X]$, I want to derive an algorithm which computes a factorization $$f=\prod_{i=1}^kf_i^i\tag{1}$$ with square-free polynomials $f_i$. My Ideas: If $f'=0$, we're done ...
2
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0answers
58 views

How to prove that all primitive polynomials are irreducible

Let $F$ be a finite field, and $F[X]$ set of all polynomials in $F$, how to prove that: why all primitive polynomials $\;$ $f \in F[X]$ $\;$ must be an irreducible. Note: Polynomial primitive is an ...
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1answer
51 views

Binomial Congruence (mod 5) Identity

I've got a (hard?) Putnam-style problem that I've been given to look at . . . I've never worked any problem even vaguely like this, but my director thinks I should be able to do it. I doubt it (100% ...
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1answer
49 views

Proving degree $n$ have at last $n$ roots in $F_q[X]$

How to prove that in $F_{q}[X]$ of degree $n$, have $n$ roots?