All Questions
11 questions
9
votes
1
answer
782
views
Is every field the residue field of a discretely valued field of characteristic 0?
Let $k$ be a field of positive characteristic $p$. Is there necessarily a discrete valuation ring of characteristic $0$ with maximal ideal $(p)$ and residue field isomorphic to $k$?
- 2,130
3
votes
1
answer
136
views
Rank 1 valuations that are not discrete on finite transcendental extensions of the rationals
Suppose $K=\mathbb{Q}(X_1,\dots,X_n)$ is a purely transcendental extension of the rationals on finitely many indeterminates. Can anyone give an example of a rank $1$ valuation on $K$ that fails to be ...
- 19.6k
4
votes
1
answer
348
views
Given a non-field local domain $R$, finding a dominating Valuation ring whose residue field is algebraic/finite extension of the residue field of $R$
Let $(R, \mathfrak m)$ be a non-field local domain with fraction field $Q(R)$ . Let $k_{R}:=R/m$.
We know that there is a Valuation ring $(V,\mathfrak m_V)$ such that $R \subseteq V \subsetneq Q(R)$ ...
user111524
3
votes
0
answers
96
views
Luroth's theorem for Discrete valuation rings?
Luroth's theorem states that if $k$ is a field and $L$ is a field extension of $k$ such that $k \subset L \subseteq k(X)$, then $L=k(f(X))$ for some $f(X) \in k(X) $ . My question is ; is there any ...
user111524
5
votes
0
answers
209
views
Completions of $K(x)$
Let $K$ be a field. Are there books or articles discussing completions of $K(x)$ with respect to the metric induced by the $p$-adic valuation $|\;\;|_p$ where $p\in k[x]$ is irreducible and different ...
- 596
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
0
votes
0
answers
383
views
Pseudo-cauchy sequence and valuation
Let $k$ be a field and $x$ is transcendental over $k$. Can we construct a pseudo-cauchy sequence $(a_{i})$ convergent to $x$ with each $a_{i}$ is algebraic over $k$ and $k(a_{i})\subseteq k(a_{i + 1})$...
- 173
2
votes
1
answer
601
views
Henselization of valued field
What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
- 173
1
vote
0
answers
196
views
Extending commuting endomorphims of a complete discrete valued field to the algebraic closure?
Is it true that any two commuting endomorphisms of a complete discrete valued field extend to commuting automorphisms of the algebraic closure?
- 341
8
votes
1
answer
351
views
When is a valued field second-countable?
Let $R$ be a valuation ring, with fraction field $K$ and residue field $k$. Denote by $\Gamma=K^{\times}/R^{\times}$ the valuation group (assumed nontrivial).
The valuation $v:K^{\times}\to\Gamma$ ...
- 19.6k
11
votes
2
answers
863
views
Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
- 19.6k