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Results tagged with algebraic-number-theory
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user 18238
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
5
votes
explicit uniformizer for the false Tate extension
I think the answer to question Q2 is no. Indeed, Lemma 3 of Birch's paper in Cassels and Fröhlich tells us that two extensions $k(\sqrt[n]{a}),k(\sqrt[n]{b})$ of a field $k$ (of characteristic prime t …
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2
votes
Accepted
Does the Hasse norm theorem easily imply the global squares theorem?
Well, I would say that this crucially depends on what you define to be "quick". If you admit global class field theory, at least in its idelic formulation, the fact that all primes in $K$ are split in …
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0
votes
Decomposition of primes in Galois closures of number fields
The answer is no. Let $K=\mathbb{Q}$ (it does not really matter) and let $M/\mathbb{Q}$ be a degree $6$ extension with Galois group $S_3$; finally, pick for $L$ any of the three cubic subfields of $M$ …
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3
votes
Computing certain class numbers modulo 4
I come later than GH and paul Monsky and with an intricate and way too long answer, but since I had fun in solving the exercice, I post it anyhow. Most probably (as paul Monsky suggested) there are ea …
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5
votes
Accepted
Refinement of (classical) Iwasawa main conjecture
There is an answer which follows from Kolyvagin's theory of Euler Systems, and can be found in Theorem 4.4 of
K. Rubin, Kolyvagin's System of Gauss Sums, in Arithmetic Algebraic Geometry, van der Gee …
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0
votes
Accepted
About real abelian number fields
Let $J^{(p)}\subseteq J$ be the subfield fixed by the $p$-Sylow subgroup of $\operatorname{Gal}(J/K)$ which is also the $p$-Sylow subgroup of the (abelian!) group $\operatorname{Gal}(J/\mathbb{Q})$ si …
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6
votes
Ideal classes fixed by the Galois group
As Franz Lemmermeyer suggested, one should consider the so-called Ambigous Class Number Formula: you find it, for instance, in Gras' book "Class Field Theory", II.6.2.3.
It says that if $L/K$ is a fi …
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3
votes
Accepted
About principal ideal theorem in number fields
I have been trying to prove the result for a couple of days, without success, so I post what I got in the meanwhile. Let me suppose throughout that $\operatorname{Gal}(E/K)\cong(\mathbb{Z}/p)^2$ (the …
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1
vote
Accepted
A question on Cebotarev's density theorem
Edit: I understand that you are happy with a $T$ which depends upon $L$ and works simultaneously for all $L'/L$ of degree bounded by $d$ and unramified outside of $S_L$. If you are looking for a unifo …
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1
vote
Brauer group of complete DVR
I think an even easier answer (although it should more or less boil down to Ekedahl's) is to apply snake lemma to the following diagram - noting that the middle vertical arrow is an isomorphism
$$
0\t …
- 5,670
6
votes
Commutative algebra with a view toward algebraic _number theory_
According to Mariano's request, I turn my comment into an answer:
I think Neukirch's book "Algebraic Number Theory" might be a good reference. The first part "look reasonably abstract" to be though …
6
votes
An elementary, short proof that the group of units of the ring of integers of a number field...
I do not know if the following qualifies as "short" or "elementary": but it does not follow the usual pattern through Minkowski's Convex Body Theorem. Rather, it mimics the classical proof of Mordell– …
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6
votes
Accepted
The $\ell$- part of the class groups of the $p$-cyclotomic fields
As Keith Conrad observed in his comment, what you are asking about is Washington's theorem which is either in his book, referenced as Theorem 16.12, or in his original paper in Invent. Math. He proved …
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4
votes
Accepted
Class groups in dihedral extensions - some sort of Spiegelungssatz?
I normally don't like to cite my own work on MO, but this time the preprint arXiv:1803.04064 was written, together with L. Caputo, having the OP's question in mind; and so, first of all, let me thank …
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1
vote
Accepted
Kummer congruences for totally real number fields
I think that the point lies in the difference between a primitive and imprimitive $L$-function. Before entering the details, let me observe that Washington's definition of $p$-adic $L$-functions (as t …
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