# Questions tagged [primitive-elements]

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8
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### 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**

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159 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. ...

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### 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**

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**1**answer

475 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 ...

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174 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**

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190 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 $\...

**1**

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**1**answer

336 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 ...

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589 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 $...