Questions tagged [primitive-elements]

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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(...
Mikhail Goltvanitsa's user avatar
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 $...
Andrew Clifford's user avatar
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....
aleph's user avatar
  • 503
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 $...
Bernhard Boehmler's user avatar
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(\...
P Vanchinathan's user avatar
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. ...
Mikhail Goltvanitsa's user avatar
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 $...
Angel del Rio's user avatar
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 ...
P Vanchinathan's user avatar
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}(\...
P Vanchinathan's user avatar
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 $\...
P Vanchinathan's user avatar
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 ...
Favst's user avatar
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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 $...
P Vanchinathan's user avatar