# Tagged Questions

For questions related to valuation functions on a field.

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### Discrete valuation on $\mathbb{Q}(X,Y)$ such that the residue field is $\mathbb{Q}$

I tried multiplicity of zeros and poles because this works for $\mathbb{Q}(X)$. I guessed that this would be the same for multivariable cases, but it looks more complicated and I don't know how to ...
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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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### An application of the Weak Approximation theorem - Artin-Whaples Approximation Theorem

Let us recall the weak approximation theorem from Valuation theory in Algebraic Number Theory. Let $K$ be a field, and let $|\cdot|_{1},\dots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...
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### Value group of simple field extension: Are these value groups equal?

Suppose that a field extension $L/K$ is finite, K is a Henselian field with a exponential valuation $v$, and $w$ is an extension of $v$ to $L$. (If it is necessary, we can also assume that $L/K$ is ...
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### How do we find the prime ideals of a ring of integers of a number field?

For example, for $F=\mathbb Q(\sqrt{-5})$ the ring of integers of $F$ is $\mathbb Z[\sqrt{-5}]$ (since $-5\equiv3 \pmod 4$). How can we determine the prime ideals of this ring? Another problem is the ...
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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 ...