1
vote
1answer
34 views

Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
3
votes
0answers
80 views

zeros of a simple polynomial

For $x, y \in GF[2^n]$, consider the two-parameter polynomial $P(x,y) = x \cdot y + f(x) + g(y)$, where $f$ and $g$ are arbitrary polynomials on $GF[2^n]$. Can we say anything about the number of ...
2
votes
2answers
35 views

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field.

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field. Find necessary and sufficient condition Attempt: Since, we know that a finite ...
6
votes
2answers
380 views

Let F be a field of order 32. Show that the only subfields of F are F itself and {0,1}.

$F$ is a field of order $32$. $F$ and {$0,1$} are trivial subfields of $F$. But how can we show that these are the only subfields of $F$? Can someone give me a direction to this question?
1
vote
1answer
57 views

If $X^{p^d}\equiv X\text{ mod }f$ for prime numbers $p,d$, then $f\in\mathbb{F}_p[X]$ with $\deg f=d$ is irreducible over $\mathbb{F}_p$

Let $p$ and $d$ be prime numbers and $f\in\mathbb{F}_p[X]$ with $\deg f=d$ and no roots over $\mathbb{F}_p$. I want to show $$X^{p^d}\equiv X\text{ mod }f\;\;\;\Rightarrow\;\;\;f\text{ is irreducible ...
1
vote
0answers
78 views

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 ...
2
votes
1answer
74 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 $G$?
3
votes
2answers
102 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?
2
votes
0answers
69 views

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.
3
votes
3answers
94 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?
0
votes
1answer
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 ...
2
votes
2answers
21 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 ...
3
votes
1answer
45 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 ...
2
votes
1answer
123 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 ...
1
vote
1answer
34 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?
2
votes
2answers
38 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 ...
0
votes
1answer
20 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 ...
0
votes
2answers
47 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 $a$,$b$ ...
2
votes
3answers
39 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 ...
1
vote
1answer
31 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 ...
1
vote
3answers
31 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 ...
1
vote
1answer
69 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? ...
0
votes
3answers
96 views

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 ...
0
votes
1answer
64 views

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 ...
0
votes
1answer
38 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 ...
1
vote
1answer
29 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} ...
2
votes
1answer
84 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 ...
2
votes
2answers
52 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 ...
2
votes
1answer
73 views

Having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space over $\mathbb{F}_p$.

As the title states, I'm having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space V over $\mathbb{F}_p$. Clearly it is dimension $n$. How can we work with this vector space? ...
1
vote
3answers
66 views

Show that a map is not an automorphism in an infinite field

How should I show that a map $f(x) = x^{-1}$ for $x \neq 0$ and $f(0) = 0$ is not an automorphism for an infinite field? Thanks for any hints. Kuba
1
vote
1answer
37 views

The characteristic of the field $GF(p^n)$

How to show that characteristic of the field $GF(p^n)$ is $p$? I have come across this fact on Wikipedia webpage, but don't know how to prove it. Thanks
0
votes
1answer
41 views

Discrete Logarithm Problem in $GF(p^m)$

I have question regarding DLP in $GF(p^m)$ I know the algorithms for solving the DLP in $GF(p)$ like Baby Step-Giant Step, Pohlig-Hellman etc... But what if we move into the $GF(p^m)$ and are ...
3
votes
3answers
313 views

How and in what context are polynomials considered equal? [duplicate]

There's two notions of equivalent polynomials floating around, one saying that $f = g$ iff they're equivalent as maps, and the other saying $f = g$ iff they're equal on each coefficient when written ...
1
vote
3answers
67 views

Prove that every extension of a finite field is normal

In book by Roman 'Field Theory' it is written that it is straightforward that every extension of a finite field is normal. However I just cannot see it. Can you help me with this problem? Thank
4
votes
0answers
67 views

Isomorphism between finite fields adjoining a root

Let $p(x)=x^3+x^2+1$ and $q(x)=x^3+x+1$ be polynomials over the field $\mathbb{Z}_2$. Let $\alpha$ be a root of $p(x)$ and $\beta$ be a root of $q(x)$. Now let $K=\mathbb{Z}_2(\alpha)$ and ...
0
votes
0answers
19 views

How is the table generated for Galois Field?

If I want to generate tables for $01AB\quad 01AB$ for both addition and multiplication, how will it be generated? I am basically confused from this wikipedia example! I hope someone can clear it up ...
2
votes
1answer
62 views

Structure of a module and finite fields

I am trying the following problem which appeared on an exam for a grad course: Determine the structure of finite field $F$ as a module over $F_p[x]$ when $xv=Lv$. Here $L(z)=z^p$ and ...
1
vote
2answers
32 views

Ring theory Algebra

Let F be a field of $8$ elements and A=set of all x belongs to F such that $x^7=1$ and $x^k \neq 1$ for all $k < 7$. then the number of elements in A is
0
votes
0answers
21 views

Karatsuba Method

For polynomials $f(x)$, $g(x)$ of degree $d = 2^{r-1}-1$, how do I check that multiplying $f(x)$ and $g(x)$ by the Karatsuba method requires $3^{r-1}$ multiplications in the field $F$?
1
vote
0answers
48 views

How many different proper subfields does $K$ have, where $K$ is a field of order$p^n$?

If $p$ is a prime, and $d$ is an integer greater than or equal to 1. Let $n= p^d$, then there exists $K$ being a field of order $p^n$. But how many different proper subfields does $K$ have? The ...
1
vote
2answers
71 views

Irreducible Polynomials over a Finite Field

I am motivated by this question, and want to find a solution to the following problem. Question: How many monic, irreducible polynomials of degree $n$ are there over the finite field ...
3
votes
2answers
63 views

What is an algebraically closed field of characteristic $p$?

I suspect that this is a very simple question, but I need to ask. My question is How do the fields of characteristic $p$ look like? If $K$ is a finite field of order $p^n$, then $K$ has ...
1
vote
1answer
27 views

Degree of a Finite Field

Consider the finite field of characteristic $\mathbb{F}_{p}$ and the polynomial $f(x) = x^{p^{n}}$ - x. The splitting field $f(x)$ is a field $\mathbb{F}_{p^{n}}$ with $p^{n}$ elements. Given this ...
1
vote
1answer
56 views

Why doesn't SAGE understand reduced expressions mod p in a finite field extension?

Suppose I have a finite field $\mathbb{F}_{13}$ and I would like to adjoin an element, $\zeta$, with order $3$, since $\mathbb{F}_{13}$ does not contain one. So consider $\mathbb{F}_{13}(\zeta)$. Then ...
0
votes
1answer
51 views

Isomorphism between finite fields

Refering to this question suppose I have $l(x):=x^3+x+1$ and $m(x):=x^3+x^2+1$. Then prove there is an isomorphism between $\mathbb{F}_3 [x]/l(x)$ and $\mathbb{F}_3[x]/m(x)$ I can say that elements ...
1
vote
0answers
46 views

Splitting Field of Irreducible Polynomial over a Finite Field

Suppose $f(x)$ is irreducible over $\mathbb{F}_p[X]$ and let $\alpha$ be a root of $f$ in some extension field. I want to prove the following claim. I have included thoughts below, but I am very ...
0
votes
1answer
61 views

Regarding doubt of order of element in a finite field

The problem goes as : Let $p$ be an odd prime & $\mathbb F_{p} =\mathbb Z/p\mathbb Z$.Show that: $x^{2}+1$ has a root in $\mathbb F_{p}$ iff $p \equiv 1 ( mod $ $4)$ . My Solution: $\mathbb ...
1
vote
0answers
60 views

Extension field of F2 , expressing roots and primitive elements in that field

Let $\Phi$ be an extension field of $\Bbb{F}_2$ of extension degree s >1. Let $a(x)$ be a non-zero polynomial with the coefficients in $\Bbb{F}_2$. (a) Show that if $\beta$ is a root of the ...
0
votes
1answer
44 views

How to express each element in a field F as a power of a primitive element? [closed]

I have a field F(2^4) and it is represented as a residue ring of the polynomials over F2 modulo the polynomial β4+β3+β2+β+1. I want to express each element in this field as a power of a primitive ...
5
votes
2answers
81 views

Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...