All Questions
Tagged with valuation-theory algebraic-number-theory
14 questions
0
votes
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426
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How do I extend the $2$-adic absolute value to prove Monsky's Theorem?
In proving Monsky's Theorem, it is required that we define the $2$-adic absolute value on an arbitrary finitely generated extension of $\mathbb{Q}$ say $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$...
user500557
5
votes
0
answers
119
views
Existence of invariant valuations
Given a field $K$, one can enrich it via a valuation, an automorphism or both structures at the same time in a compatible way. In all of these three cases, the model theory is well-understood (under ...
- 161
3
votes
2
answers
726
views
In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie outside the same?
This question is inspired from the post linked below:
Can an algebraic number on the unit circle have a conjugate with absolute value different from 1?
What I am curious about is the following: let $\...
- 739
3
votes
1
answer
289
views
Existence of algebraic integer with absolute value equal to reciprocal of maximum of $1$ and absolute value of a given algebraic number
Consider a number field $K$, and let $v_1, \cdots v_n$ ($n \in \mathbb N$) be some finite (i.e. non-archimedean) places of $K$. Is the following true?
For every $\alpha \in K^\times$ there exists $\...
- 739
4
votes
1
answer
418
views
Henselian valued fields for characteristic $0$: a characterization
Let $K=K(v)$ be a valued field of characteritic $0$ with non trivial valuation $v:K\rightarrow\mathbb{R}\cup\{\infty\}$. I'm looking for a proof of following characterization of Henselian property:
$...
- 5,966
1
vote
1
answer
142
views
Valuation theory on semisimple algebras used in the paper of Cohen-Martinet: reference request
I'm currently reading the paper of Henri Cohen & Jacques Martinet "Etude heuristique des groupes de classes des corps de nombres"
On the 2nd section, they recall some facts on valuations, ...
- 1,013
3
votes
0
answers
169
views
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 ...
- 2,050
5
votes
0
answers
209
views
Completions of $K(x)$
Let $K$ be a field. Are there books or articles discussing completions of $K(x)$ with respect to the metric induced by the $p$-adic valuation $|\;\;|_p$ where $p\in k[x]$ is irreducible and different ...
- 596
5
votes
2
answers
529
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The notions of $H^0(\widehat{ D})$ and $h^0(\widehat{D})$ are not satisfactory
Let $K$ be a number field with ring of integers $O_K$. Moreover consider an Arakelov divisor $\widehat{D}\in\overline{\operatorname{Div }(\operatorname {Spec }O_K)}$, namely
$$D=\sum_{\mathfrak p\;\...
- 1,237
9
votes
2
answers
2k
views
Completion and algebraic closure
Following this question:
Given a valued field $K$, denote with $\bar{K}$ its algebraic closure and with $\hat{K}$ the completion. Then both $\hat{\bar{K}}$ and $\hat{\bar{\hat{K}}}$ are complete and ...
- 1,101
3
votes
1
answer
2k
views
Completion of a finite field extension is also finite?
Let $(L,w)/(K,v)$ be a finite extension of valuation fields, and let $L_w$, $K_v$ be the respective completions of $(L,w)$, $(K,v)$. Is the field extension $L_w/K_v$ finite?
For nonarchimedean ...
- 5,169
0
votes
0
answers
724
views
Extension of a complete discrete valuation ring
My question came when I was reading the famous Tate's paper on $p$-divisible groups. At the beginning of chapter $(2.4)$ he cites this fact as obvious. If you take a complete discrete valuation ring $...
- 445
9
votes
2
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869
views
How did height in algeb. number theory/elliptic curves started?
Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people ...
- 2,275
0
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1
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481
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Relating $p$-adic Valuations of Elements in $\mathbb{C}$ and $\mathbb{C}_p$
Let $K = \mathbb{Q}(\theta)$, where $\theta$ is a root of an irreducible polynomial $g \in \mathbb{Z}[t]$. Fix a rational prime $p$. Let $\theta^{(1)}, \ldots, \theta^{(n)}$ be the roots of $g$ in $\...
- 113