Questions tagged [primitive-elements]
The primitive-elements tag has no usage guidance.
12
questions
0
votes
1
answer
231
views
Product of subspace and its inverse
$\DeclareMathOperator\GF{GF}$Let $R=\GF(q)$ be a finite field with $q=p^r$ elements, where $p$ is a prime number, $S=\GF(q^n)$ be an extension of $R$, where $n\in \mathbb{N}$, $n\geq 2$ and let $K=\GF(...
7
votes
1
answer
233
views
Primitive elements in a free group with trivial projection
For a free group $F$, an element $w$ is primitive if it is part of some free basis for $F$.
Let $\pi:F[x_0,x_1,...,x_n]\rightarrow F[x_1,x_2,...,x_n]$ be defined $\pi (x_0)=1$ and $\pi (x_i)=x_i$ for $...
4
votes
1
answer
262
views
Discrete logarithms and primitive elements in finite fields
The recent papers:
R. Granger, T. Kleinjung, J. Zumbragel, "On the Discrete Logarithm
Problem in Finite Fields of Fixed Characteristic," Trans. Amer. Math.
Soc., 370(5) (2018), 3129–3145.
T....
3
votes
0
answers
297
views
How to go from primitive idempotents in $\text{End}_A(M)$ to primitive idempotents in $A$?
Let $K$ be a large enough finite field, let $A$ be a finite-dimensional $K$-algebra.
Moreover, let $M$ be a finitely generated $A$-module and let $M = M_1\oplus ... \oplus M_n$ be a decomposition of $...
1
vote
0
answers
45
views
Polynomials evaluating to primitive elements
Let $K$ be an algebraic number field generated over the rational by some number $\alpha$, of degree $d$.
I'd like to know and understand the the set of polynomials $f(X)\in \mathbf{Q}[X]$ with $f(\...
5
votes
1
answer
203
views
Is it true that every subspace contain a primitive element?
Let $R = GF(q), q = p^r$, be a field with identity $e$, where $p$ is a prime number. Let $S=GF(q^n)$ be an extension of $R, n\geq 2$ and $K = GF(q^{mn})$ be an extension of $S$, where $m$ is prime. ...
0
votes
0
answers
139
views
Polynomial generated with primitive element modulo p
This question is equivalent to the question "Normal basis in cyclotomic number fields" that I asked recently. I am posing this question because maybe in this format somebody can have an answer:
Let $...
6
votes
1
answer
580
views
Do all algebraic number fields arise from Eisenstein polynomials?
This question came up while going through the application of Eisenstein criterion: The $p$-th cyclotomic polynomial after changing the variable $x$ to $(x+1)$ satisfies Eisenstein criterion. That is ...
3
votes
2
answers
209
views
Polynomials giving Lower Degree Elements in an Algebraic Number Field
My earlier related question
Lower Degree Elements in an Algebraic Number Field
has been given a clean answer for the first part. My present question is below:
Take a number field $K=\mathbf{Q}(\...
3
votes
1
answer
254
views
Lower Degree Elements in an Algebraic Number Field
Fix an algebraic integer $\alpha$ of degree $n$
such that the extension $K=\mathbf{Q}(\alpha)/\mathbf{Q}$ has intermediate fields.
(We can assume $K$ is Galois with non-simple Galois group.)
This $\...
3
votes
1
answer
489
views
Distribution of the powers of a primitive element of a finite field
What are known results regarding the distribution of the powers of a primitive element (generator of the multiplicative group) of a finite field? Specifically, compare the ordered list of ascending ...
4
votes
0
answers
689
views
Primitive Elements for $S_n$ Galois Extensions?
This is an offshoot of my other question two days ago.
How to apply Hilbert's Irreducibilty theorem?
But it is of independent interest.
Solutions of Inverse Galois Problem for a finite group $...