All Questions
Tagged with class-field-theory fields
19 questions
3
votes
3
answers
387
views
On subfields of the cyclotomic field $\mathbb{Q}(\zeta_p)$
Let $p$ be an odd prime. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $1\le k\le p-1$ be a divisor of $p-1$. Recently, when I learnt algebraic number theory, I met the following problem.
If we let
$$U_k=\{...
- 87
3
votes
1
answer
402
views
Is there a non-perfect field in which polynomials of large degree are reducible?
It is well-known that, in a real-closed field $K$, every polynomial of degree $>2$ is reducible in $K$. But in this case the characteristic of $K$ is zero.
My question is: there exists a non-...
- 41
6
votes
0
answers
513
views
Extensions of p-adic number fields
Let $p$ be a prime number and $\mathbb{Q}_p$ be the $p-$adic rational field. Let $E/\mathbb{Q}_p$ be a fixed finite extension. On this site, I define a finite extension $F/E$ to be "good" if ...
- 71
2
votes
0
answers
130
views
Can global fields be defined as certain topological fields like local fields?
It's known that local fields can be defined as a non-discrete, Hausdorff (equivalently non-indiscrete), locally compact, topological field, which is the same as non-trivial (i.e. neither discrete nor ...
- 452
0
votes
0
answers
140
views
Field of algebraic functions
We assume $K$ as a field of characteristic zero. By a field of algebraic functions of one variable over $K$ we mean a field $R$ satisfying $R=K(x,y)$ with $x$ being transcendental over $K$, and $R$ is ...
- 1
1
vote
1
answer
252
views
Theory of extensions of non-archimedian local fields
I'm searching for a recommendable reference dealing with theory of
non-Archimedean local fields where I can find proofs of the following claims about
finite extensions $L/K$ of non-Archimedean local ...
- 5,966
5
votes
1
answer
173
views
Is there a field with finitely many abelian extensions, that is neither separably closed nor real closed?
If $K$ has only finitely many Galois extensions, then $K$ must be either separably closed or real closed. Are there any other fields whose abelianizations are finite extensions (i.e. whose absolute ...
- 2,130
2
votes
0
answers
100
views
Fibers of reciprocity maps and higher dimensional analogs
Part I.
Say $K$ is a number field, $v$ is a finite place of $K$, $K_v$ the $v$-adic completion of $K$.
We have the local Artin map for every finite $v$:
$$\rho_v : K_v^{\times}\to\text{Gal}(K_v^{\...
user95222
2
votes
1
answer
462
views
Complete fields with algebraically closed residue field
I am looking for a reference where the following result is proven:
Let $k$ be an algebraically closed field. If $K$ is a complete and discretely valued field with residue field $k$. Then $K$ is one ...
- 87
4
votes
0
answers
248
views
Polynomial equations in many variables have solutions (Lang 1952 paper)
I am trying to understand the proof of the following result:
Suppose $F$ is a function field in $k$ variables over an algebraically closed field. Let $f_1,...,f_r \in F[x_1,...,x_n]$ be ...
- 87
2
votes
0
answers
604
views
Valuation topology vs modified valuation topology
Let $K$ be a field with valuation $v:K\to G\cup\{\infty\}$ where $G$ is an ordered abelian group. In section 7.62 of the book "Foundations of analysis over surreal number fields." Vol. 141. Elsevier, ...
- 596
5
votes
1
answer
898
views
p-adic expansion for elements in algebraic closure of p-adic numbers
In the following I will describe a proposal for the p-adic expansion of the elements of the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. My question is if this "conjecture" has been ...
- 596
3
votes
1
answer
159
views
Projective coordinates over a non UFD ring
Is it true that when the integers of a number field are not a UFD then not every point in projective $n$-space over that field can be given by relatively prime algebraic integer coordinates?
When a ...
- 11.2k
9
votes
1
answer
1k
views
Class groups of orders
In Cox's book "Primes of the form $x^2 + ny^2$", he proves that in a quadratic imaginary field $K$, if $\mathcal O$ is an order of conductor $f \in \mathbb Z$, we have that the class group $\mathrm{Cl}...
- 647
3
votes
2
answers
651
views
Galois cohomology of the field of Laurent series
Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?
- 3,472
4
votes
1
answer
2k
views
Fields whose embeddings into the complex numbers are invariant under complex conjugation
Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an involution on $K$ which ...
- 41
0
votes
1
answer
469
views
Finite extensions of residue fields of Henselian DVRs
Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension ...
- 2,032
7
votes
1
answer
829
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 ...
- 73
5
votes
0
answers
196
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 in $R=...
