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### Which number fields are monogenic? and related questions

A number field $K$ is said to be monogenic when $\mathcal{O}_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}_K$. What is currently known about which $K$ are monogenic? Which are not? From ...

**22**

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### Global fields: What exactly is the analogy between number fields and function fields?

Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?"
There seems to be a general philosophy that problems over function fields are easier to ...

**41**

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### Degree of sum of algebraic numbers

This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer.
Let $a$ and $b$ be algebraic numbers, with respective degrees ...

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### sum of squares in ring of integers

Lagrange proved that every (positive) rational integer is a sum of 4 squares.
Are there general results like this for ring of integers of a number field? Is this class field theory?
Explicity, ...

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### Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?

Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? I am going to be on the road soon, so pleas don't be offended if I don't respond quickly to a comment.

**11**

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### A problem on Algebraic Number Theory, Norm of Ideals

A problem on Algebraic Number Theory
$K$ and $L$ are number fields over $\Bbb Q$.($\Bbb Q$ is rational number field)
$K\subseteq L$.
$\mathcal{O}_K$ is the ring of integers of $K$.
and ...

**11**

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**2**answers

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### Which polynomials arise as formulas for a conjugate

For any integer $r \geq 2$, et $V_r$ be the set of polynomials $Q \in {\mathbb Q}[X]$ of degree $r-1$ such that there is an algebraic number $\alpha$ of degree $r$ , such that
$Q(\alpha)$ is a ...

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**0**answers

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### Normal basis in cyclotomic number fields

Let $p$ be an odd prime integer and let $\zeta$ be a primitive $2p$-th root of unity. Does $\alpha=1+\zeta+\zeta^{-1}+\dots+\zeta^{\frac{p-1}{2}}+\zeta^{-\frac{p-1}{2}}$ generates a normal basis of ...