# Tagged Questions

For questions related to valuation functions on a field.

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### Geometric interpretation of a result from commutative algebra

I have come across the following result in Hartshorne, $I.6.5$ for those who have the book. The result says that if $K$ is a finitely generated extension of some base (algebraically closed) field $k$ ...
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### Does every coset in a ring of integers contain a totally positive element?

Let $K$ be a number field with ring of integers $\mathcal O_K$, let $\mathfrak m$ be an ideal in $\mathcal O_K$ and let $a \in \mathcal O_K$ such that $(a, \mathfrak m) = 1$. Does there necessarily ...
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### Non-smooth curve in $\mathbb{A}^2$

In one of my exercises on Algebraic Geometry, I showed that the curve $X \subset \mathbb{A}^2$ defined by $x^3-y^2$ is irreducible but not smooth. Furthermore, they ask the following question that I ...
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### The Archimedean place of $\mathbb{Q}$

Is there a way to extract the Archimedean absolute value of $\mathbb{Q}$ from its field structure in a way analogous to its non-archimedean absolute values? Here is some context: Given a valuation ...
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### Does a place $v$ of a number field $K$ ramify in $L/K$ iff $v\mid d_L$?

Let $L/K$ be an extension of number fields and $v$ be a prime (an equivalence class of valuations) of $K$ and $d_L$ the absolute discriminant of $L$. I know that a rational prime $p$ in $\mathbb Q$ ...
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### non-Archimedean Valued field extension of $\mathbb{R}$

Let $K$ be a field with non-Archimedean valuation $|\cdot|$. Suppose that $\mathbb{R}\subset K$. Question 1: Is the restriction of $|\cdot|$ to $\mathbb{R}$ the trivial valuation? I guess that the ...
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### Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
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### Does every non-Archimedean absolute value satisfy the ultrametric inequality?

The Archimedean property occurs in various areas of mathematics; for instance it is defined for ordered groups, ordered fields, partially ordered vector spaces and normed fields. In each of these ...
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### Valuation rings of dim 1,2

I am studying valuation rings (beginner). I have read some theorems but still don't know a nontrivial example. Please give me an example which is not field. Also Need help to have examples of Krull ...
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### $p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
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