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Results tagged with algebraic-number-theory
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user 35575
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
3
votes
Accepted
Lower bound on the class group of the p-Hilbert class field of an imaginary quadr. field
If $G$ is the Galois group of the $p$-class field tower over $K$, then $A(H(K))=G'/G''$ is a quotient of $G_2/G_4$, where $G_i$ denotes the lower central series. By Arrigoni's calculation that $G_2/G …
- 8,467
2
votes
relations between class numbers of quadratic extensions
The short answer to your question is basically no, there's essentially no connection between the prime powers $q^i$ dividing $h_p$ and $h_{-p}$.
It's true that there's a general relationship betwee …
- 8,467
17
votes
Conceptualizing Weil Pairing for elliptic curves ( and number fields)
The unifying picture you're looking for is probably most transparent the other way around -- by re-writing the Weil pairing on elliptic curves (in fact, this works more generally for Jacobians) to mak …
- 8,467
5
votes
Accepted
Congruences mod primes in Galois extensions
Sure. $a\equiv b\pmod{\mathfrak{P}}$ just means $a-b\in\mathfrak{P}$. Taking norms to any subfield $K$ of $\mathbb{Q}(\zeta_n)$ (e.g., $\mathbb{Q}$ or $\mathbb{Q}(\zeta_m)$) gives you $N_{\mathbb{Q} …
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12
votes
Where can I find a modern write-up of Heegner's solution of Gauss' class number 1 problem?
In his article On the "gap'' in a theorem of Heegner, Stark does a pretty thorough job of explaining where people thought the purported gap came from, to what extent it actually was a gap, and what yo …
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16
votes
Commutative algebra with a view toward algebraic _number theory_
I concur that Neukirch is a good candidate, so instead of starting with a new recommendation (I'll come back to that later), let me instead disagree with Felipe Voloch's contention that algebraic numb …
26
votes
1
answer
2k
views
What is the ring of integers of the Pythagorean field?
Following Hilbert, we call the complex numbers constructible via
compass and straight-edge the field of Euclidean numbers, and
the totally real such numbers the field of Pythagorean numbers. (Among o …
- 8,467
47
votes
3
answers
5k
views
Class Numbers and 163
This is a bit fluffier of a question than I usually aim for, so apologies in advance if this doesn't pass the smell test for suitability.
Likely my favorite fun fact in all of number theory is the jux …
- 8,467
73
votes
2
answers
10k
views
Please check my 6-line proof of Fermat's Last Theorem.
Kidding, kidding. But I do have a question about an $n$-line outline of a proof of the first case of FLT, with $n$ relatively small.
Here's a result of Eichler (remark after Theorem 6.23 in Washingt …
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5
votes
Accepted
Etale coverings of certain open subschemes in Spec O_K
As Kevin points out, $V$ is indeed $\mathcal{O}_K[\frac{1}{2}]$ in your example. Your link to the fundamental group is also correct. $\pi_1(U)$ is the Galois group of the maximal extension of $\math …
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7
votes
On what kind of objects do the Galois groups act?
This is not exactly an incarnation of the question you asked, in the sense that is not so much an action of a Galois group but rather an action whose existence is governed by a Galois group of number- …
- 8,467
31
votes
3
answers
3k
views
Why aren't there more classifying spaces in number theory?
Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested in...) …
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39
votes
Good algebraic number theory books
Though Mariano's comment above is no doubt true and the most complete answer you'll get, there are a couple of texts that stand apart in my mind from the slew of textbooks with the generic title "Alge …