# Questions tagged [global-fields]

The global-fields tag has no usage guidance.

The global-fields tag has no usage guidance.

27
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Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $...

2
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0
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49
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Suppose $K$ is the field of fractions of a Dedekind domain $R$, and let $L$ be a finite extension of $K$. There is a notion of ramification of primes of $K$ in $L$, which describes how $\mathfrak p \...

1
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0
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135
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I was wondering which computer algebra programs out there can solve polynomial systems on the algebraic closure of $\mathbb{Q}$ analytically and efficiently.
So far, I only found MAGMA with its ...

2
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0
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115
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One way to formulate local class field theory is by saying that the local Artin map induces an isomorphism from the profinite completion of $K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$, which ...

1
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0
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I am reading the paper ''Endoscopic classification of representations of quasi-split unitary groups'' by Chung Pang Mok, and cannot come up with the proof of theorem 7.2.1.
Here is the statement.
...

1
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0
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49
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Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...

2
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0
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74
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There are two concepts of self dual character, one is for global and another is for local.
Let $K$ be an imaginary quadratic number field, and a Hecke character $\chi : \mathbb{A}_K^{\times}/K^{\...

4
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1
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687
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Let $K$ be a global field (number field or algebraic function field over a finite field), $\mathcal{V}$ the set of $\mathbb{Z}$-valuations on $K$, $S \subseteq \mathcal{V}$ a finite set. The ring of $...

2
votes

1
answer

124
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Let $p$ be a prime, $f\in \overline{\mathbb F}_p[x]$ a polynomial of degree $>1$ and $t$ be transcendental over $\mathbb F_p$. Let $i\geq 0$ and let $M=\overline{\mathbb F}_p(t)(\alpha)$, where $\...

1
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0
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Consider the quadratic forms
$$
Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2
$$
over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...

4
votes

1
answer

195
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There is a result due to Siegel that, for a number field $K$, any totally positive element of $K$ is the sum of four squares of $K$. This is discussed in another question (sum of squares in ring of ...

3
votes

1
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219
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Let $F=\mathbb F_{2^n}(t)$, and let $f=x^2+ax+b\in F[x]$. Is there any necessary and sufficient condition for $f$, depending on its coefficients, to have a root in $F$? I'm not interested in finding ...

5
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2
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239
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Let $K$ be a global field and $\upsilon$ a place of $K$. Let $K_{\upsilon}$ denote the completion of $K$ at $\upsilon$ and $K_{(\upsilon)}:=K^{sep}\cap K_{\upsilon}$ the henselization (which is the ...

6
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2
answers

701
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I asked this question on math.stackexchange but maybe it fits here better. If not, I apologize in advance and will remove the question.
Let $K$ be a global field and $\upsilon$ a prime of $K$. Then ...

1
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0
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140
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Let $K/\mathbb{F}_{q}(X)$ be a global function field extension of degree $2$, having minimal polynomial $Y^{2}=f(X)$, where $f(X)$ is a degree 3 polynomial in $X$ with coefficients in $\mathbb{F}_{q}$,...

4
votes

1
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It is well known that every local field (i.e. nondiscrete topological field locally compact with respect to the topology) is the completion of some global field. I know the argument, a nice ...

2
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0
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Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (https://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn'...

4
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0
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I posted the same question earlier in stack exchange,
(https://math.stackexchange.com/questions/1130391/algebraic-proof-of-2nd-inequality-of-global-class-field)
thinking it is most definitely not a ...

5
votes

1
answer

262
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Let $K$ be a global field, $\mathbb{A}_K$ the ring of adeles, and $U$ a unipotent algebraic group over $K$. Why is $U(\mathbb{A}_K)/U(K)$, when endowed with the quotient topology, compact?

2
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0
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386
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Let $K=\mathbb F_q((t)), p -$ prime ideal in $K$, $\psi_p$ be the local Artin map$K_p^* \to Gal(K_p^{ab}/K_p)=G_p \subset Gal(K^{ab}/K)$. Then I define global Artin map $\psi_K$as product of $\psi_p$, ...

9
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3
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I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also ...

69
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3
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The background is as follows: I have been whittling away at my commutative algebra notes (or, rather at commutative algebra itself, I suppose) recently for the occasion of a course I will be teaching ...

17
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4
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If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following:
$$0\to Br(F)\to\oplus_v Br(F_v)\to\mathbf{Q}/\mathbf{Z}\to 0.$$
The ...

6
votes

1
answer

579
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Global fields consist of finite extensions of $\mathbb{Q}$ (algebraic number fields) and finite extensions of $\mathbb{F}_q(x)$ (function fields in 1 variable over a finite field). The latter are ...

58
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9
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I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about ...

12
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2
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This is a follow-up to the following answer:
Solvable class field theory
in which it is stated as a "folklore" conjecture that the maximal solvable extension of Q is pseudo algebraically closed (...

29
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5
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Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?"
There seems to be a general philosophy that problems over function fields are easier to ...