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### 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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### 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 ...

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### 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 ...

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### 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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### 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 ...