2
votes
0answers
71 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.
0
votes
2answers
38 views

Subgroups of Galois groups of finite fields

According to the notion of Galois group, for $E=GF(2^n)$ as an extension of the field $F=GF(2)$, the Galois group $Gal(E/F)$ is a cyclic group of order $n$. Now my question is: for finding the ...
2
votes
3answers
100 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 ...
2
votes
0answers
62 views

How to prove that if $f(x)$ is primitive over $GF(2^m)$ then its reciprocal is primitive too? [closed]

How to prove that if $f(x)$ is primitive over $\mathbb{GF}(2^m)$ then its reciprocal $f^*(x)$ is primitive too?
2
votes
1answer
49 views

if $\operatorname{ord}(\alpha)=n_1$ and $\operatorname{ord}(\beta)=n_2$, then what is $\operatorname{ord}(\alpha\beta)$?

$\newcommand{\ord}{\operatorname{ord}}$ If $\alpha,\beta \in \mathbb{GF}(q)$ and $\ord(\alpha)=n_1$ and $\ord(\beta)=n_2$, then what is $\ord(\alpha\beta)$ ? Edit: if $k=\ord(\alpha\beta)$ then ...
0
votes
0answers
286 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 ...
3
votes
1answer
325 views

Prove that the group, under multiplication, of all nonzero elements in a finite field must be a cyclic group.

Prove that the group, under multiplication, of all nonzero elements in a finite field must be a cyclic group. This is what I did, but I'm not sure if it's right: First, we look at the additive ...
1
vote
1answer
179 views

Non-trivial 93rd roots of unity in finite fields [duplicate]

Possible Duplicate: Finding the values of $n$ for which $\mathbb{F}_{5^{n}}$, the finite field with $5^{n}$ elements, contains a non-trivial $93$rd root of unity For which of the following ...
2
votes
2answers
284 views

Generators of Finite Fields and Quadratic Extensions

I want to show that if an element in $\beta \in K$ where $|K|=p^n$ is a generator for $K^*$, i.e. has order $p^n-1$, then there is a generator $\alpha\in L$, $[L,K]=2$ i.e. of order $(p^n)^2-1$, such ...
1
vote
2answers
134 views

Cycling through powers of a generator of finite field. Something similar to modpow for Z/nZ

I am not sure if I am talking to the correct community (perhaps stack overflow is best). I want to be able to compute $$g^x = \underbrace{g \cdot g \cdot \dots \cdot g}_\text{x amount of times}$$ ...