Articles of valuation theory

Valuation rings and total order

Let $K$ be a field and $\mathcal{O}$ be a valuation ring of $K$ (So $\mathcal{O}\subset K$ is an integral domain with the property that either $x$ or $x^{-1}$ is in $\mathcal{O}$ for all $x\in K$). Define a valuation $\nu: K^\times\rightarrow K^\times/\mathcal{O}^\times$ by the quotient map, where $\cdot^\times$ denotes the group of units. Now for elements […]

Discrete valuations of the rational numbers

I’m trying to find every discrete valuation on the field of rational numbers. If $a\in \mathbb Q$, we can write $a=p^j\frac{x}{y}$, where $p$ is a prime number and $p\nmid x$ and $p\nmid y$. We can define the p-adic valuation $v_p(a)=j$. I found easy to prove this is indeed a discrete valuation. I would like to […]

Existence of an element of given orders at finitely many prime ideals of a Dedekind domain

Let $A$ be a Dedekind domain. Let $P$ be a non-zero prime ideal of $A$. Let $\alpha \in A$. Let $k$ be a non-negative integer. If $\alpha \in P^k$ and $\alpha\notin P^{k+1}$, we write $v_P(\alpha) = k$. How can we prove the following Proposition. Let $A$ be a Dedekind domain. Let $P_1,\dots, P_n$ be distinct […]

Valuations, Isomorphism, Local ring

Let $x \in \mathbb{Q}, \, x \neq 0$, such that $x=p^r \frac{a}{b}, \quad a,b, r \in \mathbb{Z}, \quad p \nmid a, \quad p\nmid b$. Let $v_p(x):=r$ and $v_p(0):= \infty$. Also, $$\mathcal O_p= \left\{ x \in \mathbb{Q} : v_p(x) \geq 0\right\}= \left\{\frac{a}{b} : p \nmid b\right\} \quad \& \quad m_p= \left\{ x \in \mathbb{Q} : v_p(x) […]

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: Does this imply that the Archimedean places of $K$ correspond bijectively to the real embeddings $K\hookrightarrow \mathbb R$ and complex […]

Is the result true when the valuation is trivial and $\dim(X)=n$?

Here I proved the following result: Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are equivalent (i.e., they induce the same topology) iff there are constants $c_1$ and $c_2$ such that $c_1p_1\leq p_2\leq c_2p_1$ And here Eric Wofsey showed, […]

Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.

There appear to be two senses of the qualifier “Archimedean” for fields. One is for ordered fields, and one is for “valued fields” (fields with an absolute value function defined). In the first case, the field is said to be “Archimedean” iff there exist no elements $x$ for which $nx < 1$ for every natural […]

How many absolute values are there?

My question is the following: Are there algebraic norms on the fields $\mathbb{R}, \mathbf{Q_p}$ ”other” than the absolute value, respectively $|\cdot|_p$? Now phrasing more precisely: If generally $F$ is a field, an algebraic norm is a map $|\cdot| : K \to [0, \infty)$ such that 1) $|x| = 0 \iff x = 0$ 2) $|xy| […]

Localization of a valuation ring at a prime is abstractly isomorphic to the original ring

Let $A$ be an integral domain with quotient field $K$. We say that $A$ is a valuation ring if for any $0 \neq x \in K$, $x$ or $\frac{1}{x}$ lies in $A$. Then $A$ is necessarily a local ring. If $A$ is a valuation ring, then any ring $B$ between $A$ and $K$ is also […]

Is the topology of the p-adic valuation to the unramfied extension discrete?

Consider $\mathbb Q_p^{\text{ur}}$ the maximal unramified extension of the p-adic numbers. Suppose that on $\mathbb Q_p$ we have the usual absolute value that extends $|\frac{a}{b}|_p=\frac{1}{p^{v_p(a)-v_p(b)}}$ to $\mathbb Q_p$. Now it is known that $|\cdot|_p$ extends uniquely to $\mathbb Q_p^{\text{ur}}$. Is this absolute value discrete w.r.t. $\mathbb Q_p^{\text{ur}}$ ? The relevant definitions can be found here […]