# Questions tagged [valuation-theory]

For questions related to valuation functions on a field.

428 questions
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### Definition of the Norm Residue Symbol

Let $K$ be a field and $v$ its valuation. Let $K_v$ denotes the completion of $K$ with respect to the valuation $v$. If $L/K$ is a finite Galois extension, then all the $L_w$, for $w$ extending $v$, ...
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### Units in a discrete valuation ring.

I'm doing problem 2.26 in the book "algebraic curves" by Fulton: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf One is given two DVR's R and S both of which have the same field of fractions $K$...
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### On $A$ algebra homomorphisms $A[[X_1,…,X_n]]\to Q(A)$, where $A$ is a complete DVR

Let $(A,\mathfrak m)$ be a complete Discrete Valuation Ring (complete w.r.t. the $\mathfrak m$-adic topology) with fraction field $K$. Let $\phi : A[[X_1,...,X_n]]\to K$ be an $A$-algebra ...
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### Desingularisation of curves (Lorenzini-Invitiation to Arithmetic Geometry, chap 6,ex 7)

Given a nonsingular complete curve over algebraically closed $\bar{k}$, which is interpreted as a field $\bar{k}(X)$ of transcendence degree 1 and its set of valuations trivial on $\bar{k}$, we may ...
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### Suppose that the spot price…

Suppose the spot price of gold is \$300 per ounce and the risk-free interest rate for one year is 5%. What is a reasonable value for the one-year forward price of gold? The answer is \$315, right? ...
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### Galois Groups in Ramification Theory

I had a slight confusion about Galois groups over a base field which is complete with respect to a discrete valuation. We know that there are irreducible polynomials such as $X^3+X^2+2X-8$ where ...
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### non commuative division ring with a discrete valuation

I do not manage to exhib a non commutative ring of division with a discrete valuation. Can anyone show me one? Examples with quaternions would be a plus !! Thanks in advance.
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### Valuation ring and localizations [closed]

Theorem. Let $D$ be an integral domain with identity. The following conditions are equivalent. (1) $D_P$ is a valuation ring for each proper prime $P$ in $D$. (2) $D_M$ is a valuation ring for each ...
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### How to find the Newton polygon of the polynomial product $\ \prod_{i=1}^{p^2} (1-iX)$
How to find the Newton polygon of the polynomial product $\ \prod_{i=1}^{p^2} (1-iX)$ ? Answer: Let $\ f(X)=\prod_{i=1}^{p^2} (1-iX)=(1-X)(1-2X) \cdots (1-pX) \cdots (1-p^2X).$ If I multiply , ...
How do I prove the fact that for any valuation ring $V$ the ideals are totally ordered under inclusion?