# Tagged Questions

For questions related to valuation functions on a field.

38 views

30 views

### Topology for the decomposition of $L\otimes_K K_\frak p$

Let $L/K$ be an extension of number fields, $\frak P|\frak p$ primes of $\mathcal O_L$ and $\mathcal O_K$, and let $K_\frak p$ be the completion of $K$ wrt $|\cdot|_\frak p$ and $L_\frak P$ the ...
47 views

### Why does discrete valuation need to be surjective?

My question appeared while I was solving a problem in Abstract Algebra of Dummit and Foote. That is problem 26. My solution is almost the same as the proof in this link. However, I'm still ...
34 views

41 views

### Separable extensions of complete valued fields are automatically totally ramified?

I come up with the following argument which seems to be too good to be true: Suppose that $L|K$ is finite separable extension of complete valued fields. Let $\nu, \nu'$ be the valuation on $L$ and $K$...
31 views

34 views

### Relationship between residue class fields between extension

Let $K$ be a field with respect to a valuation and $L$ be a finite extension. For simplicity, assume $K$ is complete. From theory, valuation on $K$ extends to $L$ and the extended valuation on $L$ is ...
15 views

### Proving that certain limits exist

Let $K$ be a discrete valuation field where $\nu:K\longrightarrow\mathbb Z$ is a surjective valuation. Let $\lambda\in]0,1[\subseteq\mathbb R$, then the valuation $\nu$ induces a metric $d_\nu$ on $K$...
16 views

### question on proof in Frohlich/Taylor regarding extension of a discrete absolute value on a complete field

I am having some trouble understanding a proof in Fröhlich and Taylor, pages 105-106. There $K$ is a complete field with a discrete absolute value $u$ $\mathfrak o,\mathfrak p$ is the valuation ring, ...
24 views

### Characterization of valuation domains by means of their maximal ideal

I know the following theorem. Let $V, W$ be two valuation domains with the same fraction field $K$. Suppose $V$ and $W$ share the same maximal ideal. Then $V=W$. In other words, fixed a field $K$...
108 views

### Approach to build multivariate ranking system?

I have data of various sellers on ecommerce platform. I am trying to compute seller ranking score based on various features, such as 1] Order fulfillment rates [numeric] 2] Order cancel rate [...
27 views

### Extending valuations and linear disjointness of fields

Let $F$ be a field and let $K$ be a field extension of $F$. Suppose that $K$ can be written as the compositum of field extensions $E$ and $L$ of $F$, linearily disjoint over $F$, thus $K$ can be ...
67 views

### Reduction map on torsion points of an elliptic curve and their valuation

Let $K$ be a field of characteristic zero, complete with respect a discrete valuation $v$. Assume that the residue field $k$ is of positive characteristic $p$. Now take an elliptic curve $E$ defined ...
52 views

45 views

### Two dimensional valuation domain with value group $\Bbb Z \oplus \Bbb Q$

Let $R$ be a two-dimensional valuation domain with prime ideals $0 \subset P \subset M$ and value group $G=\Bbb Z \oplus \Bbb Q$. Then $M^2=M$ and $P^2 \neq P$. Can we say $P^n \neq P^{n+1}$ for ...
52 views

### Projective general linear group on discrete valuation ring

Let $R$ be a complete discrete valuation ring and $k$ its residue field. Let $H$ be a finite subgroup of $PGL_2(k)$ such that its order is prime with char($k$). Is there some elementary way to show ...
73 views

### Valuations on $\Bbb Q(t)$

Ex. $2.3.3$ in Algebraic Number Theory by Neukirch is the following: Let $k$ be a field and $K = k(t)$ the function field in one variable. Show that the valuations $v_{\mathfrak p}$ associated to ...
105 views

### About the usage of the strong approximation theorem

I'm reading Henning Stichtenoth's Algebraic Function Fields and Codes and at Proposition 3.2.5(a) of section 2, chapter 3 he says: Let $\mathcal{O}_S$ be a holomorphy ring of $F/K$. Then $F$ is ...
27 views

### Question on valuation axioms - Relating to $\mathbb{R}$

In an earlier thread, I asked if there was a standard generalization of the absolute value of $\mathbb{C}$ that could be placed on a field, but might not take values in $R_{\geq 0}$. What somebody ...
40 views

106 views

### A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
52 views

### Extension of $|\cdot|_\infty$ on $\mathbb R$ to $\mathbb C$

Let $|\cdot|$ be the usual absolute value on $\mathbb C$. My question is: Is the only extension of $|\cdot|$ on $\mathbb R$ to $\mathbb C$ $|\cdot|$ itself? I'm not sure about the uniqueness. I ...
63 views

### If $\mathcal O_P(C)$ is a DVR, then $P$ is non-singular

Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$. I would like to prove if $$\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$$ is a DVR, then $P$ is non-singular, i.e., the ...
Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$ a non-singular point. I want to prove that $\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$ is a DVR. I've already proved that $\... 1answer 52 views ### Archimedean completion of a number field Let$K$be a number field. I want to show that Every Archimedean absolute value of$K$is equivalent to the absolute value$|x|:=|\sigma(x)|_\infty$for$x\in K$where$\sigma$is an embedding of ... 0answers 146 views ### Archimedean places of a number field Let$K$be a number field with an Archimedean absolute value$|\cdot |$and let$\bar{K}$be the completion of$K$wrt this valuation. Then$\bar{K}\cong \mathbb R $or$\mathbb C$. My question is: ... 0answers 93 views ### An infinite prime can ramify right? (So what is Neukirch talking about?) I have been under the impression for several years that if$L/K$is an extension of number fields, then an infinite place of$K$is said to ramify in$L$if it comes from a real embedding of$K$which ... 0answers 43 views ### Geometric structure on the set of valuation rings of a field Let$K$be a field. Let$\mathcal{O}_K$be the intersection of all valuation rings with quotient field$K$. Can someone give an example of a field$K$in which we don't have a bijection of sets:$$\... 1answer 36 views ### Is$\mathcal O^{\ast}/U^{(n)}\cong (\mathcal O/\mathfrak p)^{\ast}$as topological groups? I have the following:$K$is a field with discrete valuation$v$,$\mathcal O$its valuation ring and$\mathfrak p$the maximal ideal and$U^{(n)}=1+\mathfrak p^n$the$n$-th unit group for$n\geq 1$. ... 2answers 197 views ### How to prove that any infinite algebraic extension of a complete field is never complete? My first idea is using Baire category theorem since I thought an infinite algebraic extension should be of countable degree. However, this is wrong, according to this post. This approach may still ... 2answers 78 views ### A extending the p-adic valuation to a quadratic extension of$\mathbb{Q}_p$I'm trying to solve the following problem. Prove that, if$d \in \mathbb{Z}_p$is non-square, then$|a + b \sqrt{d}|p = |a^2 − b^2d|^{1/2}_p$, for any$a, b \in \mathbb{Q}p$, defines a non-... 1answer 49 views ### Why is the separable closure of a field in it's completion not complete. I am currently reading "Algebraic Number Theory" by Neukirch and I am a bit confused by what is here. It is on page 143, it states this:$(K,v)$is a nonarchimedian valued field and$(\hat{K},\hat{v})...
I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at \$...