All Questions
21 questions
38
votes
1
answer
2k
views
Degree 17 number fields ramified only at 2
The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...
- 20.4k
34
votes
4
answers
3k
views
$A_5$-extension of number fields unramified everywhere
So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...
- 10.9k
15
votes
5
answers
4k
views
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.
- 203
12
votes
1
answer
565
views
Parametrizing all cyclic extensions of the rational numbers of degree 5
Is there a polynomial $f(T,X) \in \mathbb{Q}(T)[X]$ in the indeterminate $X$ over the field $\mathbb{Q}(T)$ with $\mathrm{Gal}(f/\mathbb{Q}(T)) \cong \mathbb{Z}/5\mathbb{Z}$ such that for every Galois ...
- 11.3k
8
votes
3
answers
2k
views
Is there a notion of Galois extension for Z / p^2?
The above title is in fact a special case of what I want to ask.
Certainly we have a well defined notion of Galois extension for $ \mathbb{Q}_p $. The intersections of these extensions to the ring ...
- 2,824
8
votes
1
answer
802
views
Unramified extensions of quadratic fields
Let $K/\mathbb{Q}$ be quadratic and let $L/K$ be an (everywhere) unramified Galois extension. If $L/K$ is abelian, then one can show that $L/\mathbb{Q}$ is Galois (eg see here). Is $L/\mathbb{Q}$ ...
- 171
7
votes
2
answers
1k
views
Galois groups and prescribed ramification
What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can ...
- 11.3k
7
votes
0
answers
157
views
Non-abelian ray class fields for local fields
Let $K$ be a non-Archimedean local field. Then, thanks to work of Koch (when $K$ has positive characteristic) and Jannsen-Wingberg (when $K$ has characteristic zero, and odd residual characteristic) ...
- 1,116
7
votes
0
answers
205
views
Reference request: projectivity of the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$
$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this ...
6
votes
2
answers
797
views
Are the abelian absolute Galois groups of these local fields isomorphic?
For a field $F$ we denote by $F^{\mathrm{ab}}$ the compositum of all finite Galois abelian extensions of $F$.
Is $\mathrm{Gal}(\mathbb{Q}_2(\sqrt[8]{3})^{\mathrm{ab}}/\mathbb{Q}_2(\sqrt[8]{3})) \...
- 11.3k
4
votes
1
answer
189
views
Class numbers in the unramified biquadratic extensions of number fields
Let $K/k$ be an unramified biquadratic extension of number fields (i.e., $\operatorname{Gal}(K/k)\simeq V_4$), and $k_1$, $k_2$ and $k_3$ its three intermediate fields. I know, in general, we can ...
- 65
4
votes
0
answers
145
views
A normal extension of a number field of given degree that does not split over a given set of finite places
Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number.
Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\...
- 14.2k
3
votes
2
answers
558
views
Dedekind Zeta functions of Biquadratic fields
Let $F/ \mathbb{Q}$ be a biquadratic field of Galois group $C_2 \times C_2$. Then I know that the Dedekind Zeta function of $F$ can be factored into $L$-functions as; $$\zeta_F(s) = \zeta(s) L(s, \...
- 577
3
votes
1
answer
904
views
Unramified extension and class field theory
I am not sure this question is proper for this site, but there is no other places that I can get an answer. So if anyone can give an answer for this, it would be very helpful to me.
Let $F$ be a ...
- 601
3
votes
1
answer
198
views
Centralizer of the absolute Galois group of a number field
By this answer, we know that if $K/\mathbb{Q}_p$ is a finite extension, the centralizer of $G_K$ in $G_{\mathbb{Q}_p}$ is trivial. The argument there uses that the abelinization of $G_K$ is the pro-...
- 4,418
3
votes
0
answers
215
views
Global class field theory and closure of unit groups
I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
- 85
2
votes
1
answer
307
views
A question about unramified quadratic extension of number field
Is there any condition over a number field $K$ for an unramified quadratic extension of $K$ to admit an embedding into an unramified cyclic extension of degree 4 of $K$?
- 65
2
votes
0
answers
65
views
Constructing a cyclic extension $L$ with given local behavior of a global field $K$ such that $L$ is normal over a subfield $F$ of $K$
Let $F$ be a global field without real places
(that is, a function field or a totally imaginary number field).
Let $K/F$ be a cyclic extension of degree $n$.
Let $S$ be a ${\rm Gal}(K/F)$-invariant ...
- 14.2k
2
votes
0
answers
132
views
For a quadratic extension $E/K$, condition on a character $\chi:E^\times/E^{\times 2} \to C_2$ to give a $C_4$-extension $L/K$
Let $K$ be a finite extension of $\mathbb{Q}_2$, and let $E/K$ be a quadratic extension. By local class field theory, quadratic extensions $L/E$ correspond to quadratic characters $\chi:E^\times \to ...
- 411
1
vote
0
answers
58
views
Normality in a tower of cyclic extensions of global fields, as in Artin-Tate
Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field,
and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of $...
- 14.2k
1
vote
0
answers
95
views
Abelian group extensions
Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. Is there a way to see that $K(j(E), h(E[\mathfrak{p}]))/K$ is an Abelian extension for some $\mathfrak{...
- 1,283