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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Bounds for the norm of certain additive character sums

Let $q$ be a power of a prime, let $\alpha$ be a primitive element in the finite field with $q$ elements and let $\chi$ be the canonical additive character of the field. Recently I became interested ...
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Isomorphism of Groups (C.S.I.R)

Suppose $(F , +, .)$ is the finite Field with 9 elements. Let $G = (F , +)$ and H = (F \ {0}, .) denotes the underlying additive and multiplicative groups respectively, Then $ G \cong \mathbb Z_3 ...
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Approximating polynomials over finite fields

Consider a binary finite field $F = GF[2^{n}]$ with addition and multiplication denoted by $\oplus$ and $*$, respectively. Let me represent the elements of $F$ by $n$-bit strings, which means that ...
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55 views

Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
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150 views

Polynomial multiplication modulo polynomial

Suppose we are working on finite field $F_{16}$ and have pritimive polynomial $z^4+z+1$. I stuck at how to compute polynomial modulo. For example, we have $z^5+z+1$ mod $z^4+z+1$. I use the usual ...
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133 views

What is the number of distinct subgroups of the automorphism group of $\mathbf{F}_{3^{100}}$?

Let $G$ denote the group of all the automorphisms of the field $\mathbf{F}_{3^{100}}$ that consists of $3^{100}$ elements. What is the number of distinct subgroups of $G$?
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90 views

Number of points on an elliptic curve over $ \mathbb{F}_{q} $.

I have the following elliptic curve: $$ E: \quad Y^{2} = X^{3} + 1 ~ \text{over} ~ \mathbb{F}_{q}, ~ \text{where} ~ q \equiv 1 ~ (\text{mod} ~ 3). $$ I want to know the number of points on this curve. ...
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129 views

Multiplicative group of a finite field

Field $\mathbb{F}$ is finite if and only if its multiplicative group $\mathbb{F}^{\times}$ is finitely generated. The "$\Rightarrow$" implication is obvious, but how to prove the otherwise?
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35 views

Can the set L, of all even polynomials be a subspace of F[X]?

So I have the question Let F be a field and let L be the set of all polynomials f(x) element of F[X] satisfying the condition that deg(f) is even. Is L a subspace of F[X]? I would say that L is not a ...
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Let $F$ be a field. If $F^*$ is a cyclic group, must $F$ be a finite field? [duplicate]

Let $F$ be a field. If $F^*$ is a cyclic group, must $F$ be a finite field? (It's well-known that if $F$ is a finite field, $F^*$ is a cyclic group). Thank you in advanced.
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135 views

Why is the multiplicative group of a finite field cyclic? [duplicate]

Why is the multiplicative group $(K\smallsetminus\{0\},\cdot)$ of a finite field $(K,+,\cdot)$ always cyclic?
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37 views

Existence of “simple” irreducible polynomial of degree 12 in a finite field

Assume that we have a finite field $\mathbb{F}_p$, where $p$ is prime, $p \equiv 1\ (\textrm{mod}\ 4)$ and $p \equiv 1\ (\textrm{mod}\ 3)$. I was looking for irreducible polynomial in a form $X^{12} + ...
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23 views

Showing $f\in\mathbb{F}_{p^d}[X]:f'=0\Rightarrow\exists g\in\mathbb{F}_{p^d}[X]:f=g^p$

Let $\mathbb{F}_{p^d}$ denote the final field with $p^d$ elements and $\mathbb{F}_{p^d}[X]$ denote the polynomial ring in $X$ over $\mathbb{F}_{p^d}$. How can we show ...
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22 views

Elements with order 3 in group $F_{16}/\{0\}$

If you have the finite field $GF(16)$ and you define the group $GF(16)/\{0\},*$ this group is cyclic. I need to determine how many elements in this group have order 3. Of course you could just try out ...
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53 views

Elements of subfield $F_{8}$ of $F_{2}[x]/(x^{6}+x+1)$

I need to find the elements of the subfield $F_{8}$ of $F_{2}[x]/(x^{6}+x+1)$ in their standard representation. I know that $F_{2}[x]/(x^{6}+x+1)$ represents the residu classes of polynomials modulo ...
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129 views

An example concerning some fields

I was trying to understand the following example This is also why I've made some questions today on locally finite fields. I've almost understand everything (thanks also to some of you) but I ...
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48 views

Projective special linear group

What is it the minimum number of generators for $PSL(2,\, \mathbb{F}_q)$? Is it known? Is there some references I could see?
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76 views

Structure of $G$-invariant bilinear forms over (finite?) fields

I have a question about the structure of $G$-invariant bilinear forms. Let $G$ be an arbitrary finite group and $\mathbb{F}_q$ a finite field such that $2|G|$ is not divisible by the characteristic of ...
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54 views

Minimal polynomial reducible modulo every prime $p$

Suppose $K = \mathbb{Q}(\alpha)$ with $\alpha = a + b\sqrt{D_1}+c\sqrt{D_2}+d\sqrt{D_1D_2}$ with $D_1,D_2 \in \mathbb{Z}$. Prove that the minimal polynomial $m_\alpha(x)$ for $\alpha$ over ...
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25 views

fields and subspaces

Let F be a field and let V=F^F, which is a vector space over F. Let w be the set of all functions f element of V satisfying f(1)=f(-1). Is W a subspace of V? a. Has the zero vector b. closed under ...
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43 views

If p is a prime positive integer, find all subfields of GF(p)

If p is a prime positive integer, find all subfields of GF(p) This question just seems too vague.
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50 views

If $F$ is a field, show the following function is a permutation

Let $F$ be a field. Show that the function $a\rightarrow a^{-1}$ is a permutation of $F\{0_F\}$ So I know that if it is indeed a permutation, then it is one-to-one and onto. Also, For every $a$,$b$ ...
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36 views

Does the element exist in the Galois Field?

Let p be a prime positive integer and let a be an element of GF(p). Does there necessarily exist an element b of GF(p) satisfying b^2=a? So, taking a element of GF(p), can we find a b element of ...
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44 views

Some properties of elements of the finite field GF(q)

Consider the finite field $GF(q)$ and an element $w$ of order $n$ which is an $n$th root of unity in this field. Could someone give me explanation or reference to the following questions regarding ...
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35 views

Let $F$ be a field of 8 elements and $A$= {$x\in F$| $x^7$=1 and $x^k$$\neq$1 for all natural number k<1}. Then the number of elements in A is

Let $F$ be a field of $8$ elements and $A$= {$x\in F$| $x^7$=1 and $x^k$$\neq$1 for all natural number $k<1$}. Then the number of elements in $A$ is: a) 1 b) 2 c) 3 d) 6 Please give me some ...
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47 views

The existence of primitive algebraic units modulo 3

Consider the problem of computing $$\sqrt{2} \mod 3 $$ Whereas we seek a number $n$ such that $n^2 \equiv 2 \mod 3$ and furthermore it is known that both $n$ and $2n$ will satisfy this property, ...
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166 views

Why is the algebraic closure of a finite field countable?

An algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. But why is it a countable set?
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32 views

About a field of order $2^{n}$ with $n$ an odd integer and an additional property

I'm new in the world of fields (so I don't have any strong theorem at my disposal) and I've got stuck in this problem: Given a field of order $2^{n}$ with $n$ an odd integer and $a,b$ elements ...
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56 views

Cyclic subgroups of $GL_2(F_q)$

Let $F$ be a finite field with $q = p^f$ elements for $p$ a prime. I know that $G = GL_2(F_q)$ contains a cyclic group of order $q-1$. It is the set of matrices of the form $\begin{pmatrix} x & ...
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40 views

$x=y^p-y$, $y \in \mathbb{F}_{p^2}$. Prove: $x^2 \in \mathbb{F}_p$

Im practicing for an exam, can anyone help me with this? Let $p$ be prime and let $x=y^p-y$, for $y \in \mathbb{F}_{p^2}$. Prove: $x^2 \in \mathbb{F}_p$
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95 views

Primitive polynomials in LFSRs

I need help proving the following theorem. I found it many books but on every single one it says that they omit the proof because it is in every good textbook. THM Let $c(x)$ be a connection ...
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104 views

Additive character sum of primitive elements over finite fields

Let $\mathbb{F}_q$ be a finite field with $q$ elements and multiplicative generator $\alpha$, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $N$ be a divisor of $q-1$. ...
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The Galois group of the finite field's algebraic closure is not countable

I'm trying to prove that the group $\operatorname{Gal}(\bar F_p /F_p)$ is not countable. My idea is to show that in the sequence $F_p\leq F_{p^2}\leq F_{p^4} \leq \dots \leq F_{p^{2^n}} \leq\dots ...
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What does the Cayley table for $+$ in $\mathbb{C}$ look like?

Below is the Caley table for the $*$ operator, but how do I fill in the table for operator $+$? In general, given an operator $*$ acting on a set, $S$, can I turn this into a field by selecting the ...
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Equation over $\mathbb F_2$

Given equation over $\mathbb F_2$: $$x_1x_2x_3+x_4x_5x_6=0$$ It has $50$ solutions. Let $N$ be set of solutions. If we give some linear dependencies to variables we will get cosets (linear subspace + ...
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79 views

Cyclotomic Cosets and Minimal Polynomial for 45

Currently I am working on matlab in order to find Cyclotomic Cosets for 45. As 45 in not in the format of 2^m-1, matlab give me an error. I am trying to write algorithm in matlab/octave for my ...
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33 views

How many different bases in $\mathbb{Z}/p\mathbb{Z}$

Let $K = \mathbb{Z}_p$, for some prime $p$, and $\text{dim}\:V = n$. $V$ is a vector space over $K$. I need to find out how many different bases are in $V$. Now I know the answer is the product of ...
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Prove that $\alpha_{1} ^k+ \alpha_{2} ^k +…+ \alpha_{n} ^k = n$ for $k=0$ and $0$ for $k = 1,2,…,n-1$?

For $n\geq 2$ let $\alpha_{1} + \alpha_{2} +.....+ \alpha_{n} $ be all the nth roots of unity over a field and the roots are not necessarily to be distinct. So we have to prove that $\alpha_{1} ^k+ ...
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27 views

matrix representation of Frobenius map

I am in an urgent need to know what is the matrix representation of Frobenius map for finie field like GF(4). Let suppose the basis of GF(4) be {1, a+1}. We know that the frobenius map is generator of ...
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70 views

Theorem in finite fields fails in my example

I need to understand the following theorem, so i did an example. But i realized that i don't get everything in the finite field theory. can somebody check the example and say where the mistake is? ...
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Quotient rings of polynomial rings

I have come across a quite difficult question while I am studying for a test: Let $F=\Bbb Z[x]/(7,x^2-3)$. Let $u$ denote the image of $x$ under the canonical epimorphism from $\Bbb Z[x]$ to ...
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example about permutation in a finite field

I want to give an example to a corollary in my seminar, but i m not sure if it is ok. can somebody check it quickly? This is the corollary: Corollary: Let $n$ and $k$ be positive integers such ...
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Approximate a bivariate function by univariate functions over finite field

What is known about approximating a bivariate function by a sum of univariate functions e.g. approximating $a(x)+b(y)=f(xy)$ where $x,y \in \mathbb{F}_p$? I have in mind probably simplest non-trivial ...
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40 views

Extending a finite field twice

Assume we have a finite field $\mathbb F_p$, an irreducible polynomial $f(x)$ of degree $m$ over $\mathbb F_p$, and an irreducible polynomial $g(y)$ of degree $n$ over $\mathbb F_p[x]/(f(x))$. Then ...
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52 views

Set of Solutions of A Quadratic Equation with Coefficients in $\{0,1,\cdots , \ p-1\}$

I was just playing with quadratic equations and this interesting question came into my mind. Say I have a set of quadratic polynomials $S=\{f_{(b,c)}(x)=x^2+bx+c:b,c\in \{0,1,\cdots, p-1 \}\}$ where ...
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30 views

“$q$-linear envelopes” of $\mathbb{F}_p$-subspaces

Let $V$ be a vector space over an algebraically closed field $k$ of characteristic $p>0$, and denote by $V_q$ the vector space obtained from $V$ by restricting scalars to $\mathbb{F}_q$, where ...
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31 views

Prove for $a,b \in \mathbb{F}_{p^n}$, if $p(x) = x^3 + ax +b$ is irreducible, then $-4a^3 - 27b^2$ is a square in $\mathbb{F}_{p^n}$.

The problem is as the title states. We know that in this case determinant $D = -4a^3 -27b^2$, and also I know that if $G$ is the Galois group of $x^3 + ax + b$, then $$G \subset A_n \, \iff \sqrt{D} ...
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64 views

Endomorphism Ring of an Elliptic Curve over Finite Field

Let $~E:y^2=x^3+x~$ be an elliptic curve over finite field $\mathbb{F}_{5},$ I compute the trace of Frobenius is $2$($E/\mathbb{F}_{5}$ obvious is ordinary). (By the theory of CM, I know (when $E$ ...
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89 views

Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...
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2answers
61 views

Example for a corollary in finite field theory

I'm preparing a seminar and the problem is that I never had a lecture before in finite fields. So I had to learn everything by myself, and that is unfortunately not easy at all... Can anyone help me ...