# Tag Info

Accepted

### Motivating Lubin-Tate theory

Sorry I didn’t see this earlier. My memory is vague, and probably colored by subsequent events and results, but here’s how I recall things happening. Since I had read and enjoyed Lazard’s paper on ...
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### Any open Langlands Conjectures for GL_1?

The question is subject to interpretation, because the Langlands program was never a clearly delimited set of conjectures to begin with, and moreover many things were added during the following ...
• 25k
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• 141k
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### how to compute Hilbert class field of $\Bbb Q(\zeta_{23})$?

The cyclotomic field $K=\mathbb{Q}(\zeta_{23})$ contains the quadratic field $F=\mathbb{Q}(\sqrt{-23})$, and $F$ has class number $3$ (it is the first quadratic field, when those are ordered by ...
• 12.4k
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### On the history of the Artin Reciprocity Law

As GH has already remarked, the same thing happened a lot later after Taniyama and Shimura asked whether elliptic curves defined over the rationals are modular. To begin with your last question, there ...
• 30.9k

### Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?

I mention this as an answer since it is too long for comments. I do not know what the adelic proof assumes. Suppose that all but finitely many primes of $K$ split completely in $L$. Suppose $d$ is the ...
Accepted

### Galois groups and prescribed ramification

Let me first note that there is a slight ambiguity when one says "ramified only at 2". Strictly speaking, that means that the extension is unramified at every place of $\mathbb Q$ except 2, including ...
• 25k
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### How to compute with the Stark conjectures?

The main reference here is the very useful User's Guide, C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, "User's Guide to PARI / GP" (2003) Particularly the sections about bnrstark (pp. 108)...
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### Ideal classes fixed by the Galois group

I assume you want $K$ to be Galois over $\mathbb{Q}$. More generally, let $L/K$ be a Galois extension of number fields. The the class group $C_K$ of $K$ maps to $C_L^{G_{L/K}}$, the part of $C_L$ ...
• 42.7k
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### Existence of imaginary quadratic fields of class numbers coprime to $p$ with prescribed splitting behaviour of $p$

A more general version of this statement was shown by Kimura in Acta Arith. (2003). His corollary gives $\gg \sqrt{X}/\log X$ such fields ${\Bbb Q}(\sqrt{-d})$ with $d\le X$, and also allows you to ...
• 42.6k
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### Is cohomology of groups all about $H^{i}: -2\leq i\leq 2$?

As Serre seems to be fond of saying, life begins at $H^3$. The first appearance of a $3$-cocycle seems to be in Teichmüller's article Über die sogenannte nichkommutative Galoissche Theorie... in ...
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### How is class of composition of two quadratic fields is related class numbers of quadratic field?

I assume you mean $K_1=\mathbb Q(\sqrt{d_1})$. 1) If by $K$ you mean $\mathbb Q(\sqrt{d_1d_2})$, then there is no simple relation between $h$ and $h_1$ and $h_2$. 2) If by $K$ you mean the quartic ...
• 9,361
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### Artin reciprocity via Shimura varieties

If $K$ is 'everything 1 mod N' for some N, then the canonical model of $\mathbf{Q}^\times_+ \backslash \mathbf{A}^\times_{\mathrm{f}} / K$ is exactly $\mu_N / \mathbf{Q}$, the scheme of $N$-th roots ...
• 32.6k