# Questions tagged [global-fields]

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### Finiteness of wildly ramified cohomology

$\newcommand\p[1]{\left(#1\right)}\newcommand\Char{\operatorname{char}}\newcommand\Gal{\operatorname{Gal}}\newcommand\b[1]{\left\{#1\right\}}$ Let $K$ be a global field. All cohomology below is fppf-...
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### Basic question on the Langlands conjectures for $GL_n$ over global field of positive characteristic

My field is far from the Langlands conjectures. I am just trying to understand some basic ideas. At the moment I am interested in a global field $K$ of positive characteristic and the group $G=GL_n$. ...
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### Computer algebra programs that can solve polynomial systems on algebraically closed fields besides MAGMA

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 ...
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### Artin map and profinite completion of the idèles

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 ...
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### Globalization of a local field

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. ...
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### Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?

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 ...
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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 vote 0 answers 48 views ### How explicitly write a projective transformation between the conics over the univariate function field? 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 ... • 1,791 4 votes 1 answer 219 views ### Sums of squares in global fields (Reference Request) 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 ... • 477 3 votes 1 answer 238 views ### Quadratic equation over a global field of characteristic 2 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 ... • 509 5 votes 2 answers 242 views ### Does a$K_{\upsilon}$-point of a variety$V$give a point of$V$in$K_{(\upsilon)}=K^{sep}\cap K_{\upsilon}$for a global field$K$? 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 ... • 111 6 votes 2 answers 752 views ### Why is$K_{\upsilon}|K$separable for a global field$K$? 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 ... • 111 1 vote 0 answers 145 views ### Are elliptic global function fields totally imaginary? 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$\mathbb{F}_{q}$,... • 655 4 votes 1 answer 642 views ### Reference for: Every local field can be realized as the completion of a global field 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 ... • 412 2 votes 0 answers 119 views ### Property of a derivative in global field 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'... • 111 4 votes 0 answers 542 views ### Explicit description/calculation of norm group of ideles of characteristic$p$global field 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 ... • 111 5 votes 1 answer 297 views ### Compactness of adelic quotients for unipotent groups over global fields 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? • 1,093 2 votes 0 answers 415 views ### Global Artin reciprocity law from Local class field theory 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$\psi_p$, ... • 71 9 votes 3 answers 3k views ### Elliptic Curves over Global Function Fields 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 ... • 781 72 votes 3 answers 6k views ### Is there a "purely algebraic" proof of the finiteness of the class number? 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 ... • 64.3k 17 votes 4 answers 2k views ### Dimension of central simple algebra over a global field "built using class field theory". 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 ... • 40.2k 6 votes 1 answer 615 views ### Can algebraic number fields be generalized in a similar way to function fields in 1 variable over a finite field? 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 ... • 4,231 58 votes 9 answers 15k views ### Learning Class Field Theory: Local or Global First? 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 ... • 14.8k 13 votes 2 answers 643 views ### Evidence for$Q^{\operatorname{solv}}\$ being pseudo-algebraically-closed

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 (...