# Tagged Questions

**0**

votes

**0**answers

17 views

### Some Galois theory [migrated]

I have a question on field extensions, and I can't seem to find precise
answers when browsing through online notes etc.
Here it is: suppose $K$ and $k$ are fields with $k \leq K$ and $[K : k] = m$
...

**1**

vote

**0**answers

100 views

### Discriminant of a compositum of number fields, a bound?

Given two number fields $E$ and $F$, is there a bound on $|d_{EF}|$, the absolute value of the absolute discriminant of the compositum of fields $EF$, in terms of $d_E$, $d_F$, and the extension ...

**6**

votes

**1**answer

424 views

### Parity of class number of pure cubic fields

A pure cubic field is an algebraic number field of the form $K = \mathbb{Q}(\theta)$ with $\theta^3 = m$, $m \neq \pm 1$.
What can be said about the parity (odd or even) of the class number of a pure ...

**7**

votes

**2**answers

471 views

### An alternative description of K^*/Nm(L^*)

Is there a nice explicit description for the group $K^*/Nm_{L/K}(L^*)$ for a finite field extension $L/K$?
What if for example, $L$ is obtained from $K$ by ajoining an n-th root of some $\alpha \in ...

**3**

votes

**1**answer

684 views

### How do I visualize finite covers of curves over non-algebraically closed fields?

If $L$ is algebraically closed, fields of transcendence degree one over $L$ correspond to algebraic curves over $L$ up to birational equivalence, and finite extensions correspond to finite Galois ...

**18**

votes

**1**answer

865 views

### What is the ring of integers of the Pythagorean field?

Following Hilbert, we call the complex numbers constructible via
compass and straight-edge the field of Euclidean numbers, and
the totally real such numbers the field of Pythagorean numbers. (Among ...

**3**

votes

**0**answers

131 views

### Analogue of a ring extension splitting in the Kummer case

Background (the Kummer extension case)
Let $R$ be a complete regular local ring (it follws that it's a UFD) with a prime integer $p$ contained in the maximal ideal of $R$ (I'm mostly interested ...

**2**

votes

**1**answer

394 views

### Sum of n-th roots is rarely rational

Let $m,n$ be positive integers, and $\displaystyle \Phi_{m,n}~:~ {\mathbb{R}_+^*}^m \to \mathbb{R}_+^*, \ \ \ (x_1,x_2, \ldots , x_m) \mapsto \sum_{k=1}^m \sqrt[n]{x_k}$.
Clearly for $m=1$ if for all ...