0
votes
2answers
59 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 ...
1
vote
0answers
19 views

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 ...
2
votes
1answer
126 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 ...
2
votes
1answer
50 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 & ...
-1
votes
2answers
82 views

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 ...
2
votes
1answer
37 views

Non-obvious deduction regarding conjugates in $\text{GL}_2(\mathbb{F}_p)$

Let $\text{GL}_2(\mathbb{F}_p)$ act on $\mathbb{F}_p^2,$ the set of $2$-vectors with entries in $\mathbb{F}_p$, by matrix multiplication. $[\,$Prove that$\,]$ for any ...
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
1
vote
0answers
40 views

Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F} $

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
2
votes
2answers
54 views

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of $F$ have a square root. (Hint: For some integer $i$, ($1\leq i \leq s-1$), $i$ is odd if and only if ...
3
votes
1answer
54 views

Simpler method to show $k(x)$ is NOT a primitive irreducible polynomial?

Let $k(x) = x^2+2x+2$ be under $\mathbb Z_{11}[x]$. Determine if $k(x)$ is irreducible, and if so, determine if it is primitive. Ok, I showed that $k(x)$ is irreducible since it it a quadratic and ...
3
votes
2answers
80 views

Group-Isomorphism problem

I want to find an group-isomorphism $$ \psi : (\mathbb{Z}/8\mathbb{Z},+) \longrightarrow \mathbb{F}_9^\times $$ which should be used to multiply elements in $\mathbb{F}_9$ or to find the inverse ...
0
votes
1answer
78 views

elementary abelian groups and finite fields

in group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order $p$, where $p$ is a prime; it is a particular kind of $p$-group. now suppose that we ...
3
votes
1answer
61 views

automorphism of fields

let we consider $GF(p^n)$ as a vector space over $GF(p)$, $p$ is prime. Also we want to have an invertible linear map on $GF(p^n)$, (automorphism of $GF(p^n)$). on the other hand we know that A field ...
2
votes
3answers
101 views

Number of elements which are cubes/higher powers in a finite field.

This question is a slight generalization of This Question. How many elements are there in a finite field of order $q$ which are : Squares. Cubes. Higher powers. I mean : How many elements are ...
1
vote
3answers
93 views

For a finite field $F$ of order $n$, all elements are roots of $x^n - x$

I need to prove two things at $F[X]$ but don't know how, Ill glad to get help... $F$ is a finite field. $|F|=n$. We look at $p(x)=x^n-x\in F[X]$ 1. How we can show that every $c\in F$ is a root of ...
1
vote
2answers
43 views

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 ...
2
votes
2answers
525 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 ...
1
vote
3answers
56 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 ...
2
votes
4answers
137 views

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 ...
0
votes
1answer
161 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 ...
1
vote
1answer
70 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 ...
0
votes
0answers
56 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? ...
1
vote
1answer
51 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 ...
2
votes
0answers
186 views

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 ...
6
votes
1answer
175 views

Symmetric groups and the “field with one element”

I have heard several times that one may regard the symmetric group on $n$ letters as the general linear group in dimension $n$ over the "field with one element". In particular this heuristic would ...
0
votes
0answers
149 views

Order of orthogonal groups over finite field

The wikipedia article gives a formula for calculating the order of an orthogonal group over finite filed $O(n,q)$: I don't see how I can get such formula. Can one come up with some references?
2
votes
1answer
59 views

For what prime $p$ is $x^2=-1\pmod{p}$ solvable?

This is essentially the same as the following question: When $F_p[x]/(x^2+1)$ is a field? I don't know much about number theory. I came up with such question when I doing the following exercise: ...
0
votes
0answers
287 views

Is the multiplicative group of non-zero elements of a finite field cyclic? What are the generators of $(Z_p \setminus \{0\}$ for a prime $p$?

How to decide if the multiplicative group of non-zero elements of a finite field is cyclic or not? Based on our experience with $(Z_7 \setminus \{0\}, \cdot)$, which is generated by $3$ or $5$, and ...
6
votes
3answers
242 views

The group of invertible elements of $\mathbb F_{p}[x]/(x^m)$ is not a cyclic group.

I am stuck in a question about finite fields and would like to ask you for some help. Given an integer $m\geq 2$ and $p$ a prime number, show that $(\mathbb F_{p}[x]/(x^m))^{\times}$ (the group ...
4
votes
1answer
113 views

Conjugacy classes and orders of matrices.

The following are prime decompositions in $\Bbb{Z}_7[x]$: $x^8+1= (x^2-x-1)(x^2+x-1)(x^2+3x-1)(x^2+4x-1)$ $x^4+1= (x^2+3x+1)(x^2+4x+1)$ (a) Give representatives for the conjugacy classes of ...
0
votes
0answers
46 views

How many irreducible factors of grade $6$ there is in $\mathbb{F}_{2}\left[ x\right]$? [duplicate]

How many irreducible factors of grade $6$ there is in the polynomial ring $\mathbb{F}_{2}\left[ x\right]$? I have solved this by using the fact that every irreducible polynomial of grad $i$ is a ...
2
votes
1answer
122 views

Clarifications on the faithful irreducible representations of the dihedral groups over finite fields.

I would like to clarify a few things in the answer to this question: Faithful irreducible representations of cyclic and dihedral groups over finite fields 1) When a representation extends, what does ...
4
votes
3answers
109 views

Extending and inducing irreducible representations

I sense this may be a simple question, but it is one I haven't been able to find an answer for, possibly due to the use of different terminology. Referring to this question: Faithful irreducible ...
2
votes
1answer
169 views

Relationship between number of conjugacy classes and number of irreducible representations of a group

For a finite group G the number of irreducible representations over an algebraically closed field F is at most the number of conjugacy classes whose sizes are coprime to the characteristic of F. What ...
3
votes
1answer
256 views

Degrees of faithful irreducible representations of $\mathbb{Z}_n$ over finite fields

I came across the following theorem while studying representation theory over finite fields. A cyclic group $\mathbb{Z}_n$ has a faithful irreducible representation of degree $d$ over $\mathbb{F}_p$ ...
2
votes
3answers
79 views

Show that A is a field which has $9$ elements

For $A=\left\{\left( {\begin{array}{*{20}{c}} a&b\\ { - b}&a \end{array}} \right) | a,b\in\mathbb{Z/3Z}\right\}$ . Show that A is a field which has $9$ elements . $(A^*, .)$ is a cyclic group ...
9
votes
0answers
126 views

Subgroup structure of $\mathrm{SL}(2, p^2)$ and its irreducible characters

I am taking a course in representation theory of finite groups, and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and ...
7
votes
2answers
267 views

Exponent of $GL(n,q)$.

Another exponent problem. $GL(n,q)$ is the group of invertible $n\times n$ matrices over the finite field $GF(q)$, where $q$ is a prime power. I am trying to figure out the exponent of this group. ...
4
votes
1answer
71 views

Abelian groups related to $(\mathbb{Z}/p\mathbb{Z})^*$, with order close to $p$ — analogous to elliptic curve groups, but simpler

I'm looking for a construction of a group $H$ that is "a sister" to $(\mathbb{Z}/p\mathbb{Z})^*$, in the following rough sense: Each element of $H$ can be represented by one or a few elements of ...
6
votes
1answer
146 views

What does this notation mean: $\displaystyle\lim_{\leftarrow} \,\mathbb{Z}/n\mathbb{Z}$?

Just a small notation question from this Wikipedia page: The absolute Galois group of a finite field $K$ is isomorphic to the group $$\hat{\mathbb{Z}}=\lim_{\leftarrow} \mathbb{Z}/n\mathbb{Z}.$$ ...
2
votes
1answer
88 views

Latin squares of even order with all cells only participating in one subsquare.

For even ordered Latin squares, we can create squares in which every cell participates in a $2\times 2$ subsquare by using a simple circulant as for $n=6$ below: ...
1
vote
1answer
820 views

Irreducible representations of a cyclic group over a field of prime order

Consider $G$ a cyclic group of order $n$ with prime $p\not|n$. How do I construct all the irreducible representations over $\mathbb F_p$? How many irreducible representations are there and what are ...
2
votes
1answer
419 views

Formal derivatives over finite fields.

I am slightly confused about what formal derivatives over finite fields mean. Example 1: Consider $f(x)=x^3-2\in \mathbb{F}_7[x]$. By checking each element of $\mathbb{F}_7$ we easily see that this ...
5
votes
3answers
472 views

Other ways to deduce Cyclicity of the Units of certain groups?

The group of units of the rings $\text{GF}(p^r)$ and $\mathbb{Z}/p^r\mathbb{Z}$ are both cyclic (except for the exception of prime powers are not cyclic when $p=2$ and $r\ge 3$). This is a strong ...
2
votes
1answer
121 views

Extracting the value of $y$ from $x$ in an elliptic curve over a finite field

Given an elliptic curve $y^2 = x^3 + ax + b$ over a finite field $\mathbf{F}_p$, how can I retrieve the value of $y$ given the value of $x$? My knowledge in this area is quite limited, so I ...