Questions tagged [valuation-theory]

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Interpretation of model theory in algebraic geometry

I found a paper Some applications of a model theoretic fact to (semi-) algebraic geometry by Lou van den Dries. In this paper, the author uses model theoretical methods to prove the completeness of ...
• 427
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For an element in the integral closure of an ideal $I$ - which power is in $I$?

Consider an ideal $I$ in a ring $R$. If $f \in R$ belongs to the integral closure of $I$, then there is $k_0 \geq 0$ such that $f^k \in I^{k-k_0}$ for all $k \geq k_0$. Are there any known upper ...
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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 ...
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Is the special case of Abhyankar's lemma is also considered as such?

Consider the following statement: Assume $E$ and $F$ are unramified (over some fixed prime) finite separable extensions of a field $K$. Then $EF$ is also unramified. I always thought that it is ...
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Valued fields with quantifier elimination in the Macintyre language

For which fields $k$ of characteristic $p$ does the Witt construction of a discretely valued field $W(k)$ of characteristic $0$ with residue field $k$ eliminate quantifiers in the language of rings ...
• 2,080
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Rational power series and extensions

Let $F$ be a field, let $F(x)$ the field of rational functions, and let $F((x))$ the field of Laurent series (which contains $F(x)$). One may ask: which series $\sum_i a_i x^i$ lie in $F(x)$? The ...
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