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Results tagged with algebraic-number-theory
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user 43108
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
Non-negative integer solutions of x^2+y^3=n
As mentioned in the comments this is essentially the classic problem of finding integer points of the Mordell curve, and a lot of work has gone into it (for example towards bounding the number of solu …
- 17.6k
5
votes
Accepted
Group laws in class field theory
As mentioned in the comments, this is precisely Hilbert's twelfth problem, for the simple reason that any solution to that problem can be turned into a "group law argument" (or any solution to it is a …
- 17.6k
7
votes
An elementary, short proof that the group of units of the ring of integers of a number field...
This part of Dirichlet's unit theorem is the only one needed in the standard proof of the Weil-Mordell theorem over number fields, so you probably can find it in a few introductory books on elliptic c …
- 17.6k
7
votes
0
answers
451
views
Definition of p-adic regulator and Minkowski theory
In "Cohomology of number fields" there is an interesting analogy between the Leopoldt conjecture and Minkowski's proof of the Dirichlet's unit theorem, in particular the homomorphism
$$j_\infty: K^\t …
- 17.6k
0
votes
Effective Lindemann–Weierstrass theorem
Actually Baker's theorem generalizes Lindemann–Weierstrass, so that alredy gives you an effective bound
$$\bigg|\sum_i \beta_i e^{\alpha_i}\bigg| > Ce^{-(\log H)^k}$$
with $C$ an effectively computa …
- 17.6k
1
vote
Subfields of $\mathbb{Q}\bigl(\sqrt[n]{a}\bigr)$ for $a>0$
Regular Galois theory is more than enough for this, see the answers to this question.
- 17.6k
2
votes
Ramification of prime ideal in Kummer extension
This follows from basic properties of Kummer extensions and the Eisenstein polynomial of your prime ideal $\mathfrak{p}$.
You can find it for example in Koch's book on Algebraic Number Theory.
If yo …
- 17.6k
3
votes
Accepted
References for $K_{4k}(\mathbb{Z})$
Well, the consensus seems to be that this is an open problem, for $k >1$.
This is a quote from A. Raghuram's paper on the volume "The Bloch–Kato Conjecture for the Riemann Zeta Function" (page 8), pu …
- 17.6k
3
votes
The best possible density in Hilbert's Irreducibility Theorem
The only improvements seems to be $O(N^{s-1+|G/K|^{-1}}\log (N))$, avaible only in two cases:
The number field is $\mathbb{Q}$ (Castillo & Dietmann, 2016)
The Galois group is "small", in the sense …
- 17.6k
2
votes
Uniformity in the error term of counting ideals up to a certain norm
If $\zeta_K(s)$ has no zero for $\sigma \geq \rho$, you can take $O(x^{\rho+\epsilon})$.
This is proved for example in Lang's "Algebraic Number Theory" [XIII, §5, Theorem 6].
Even the simplest zero …
- 17.6k
8
votes
Why is Kronecker's Jugendtraum only for abelian extensions?
The bottom line is that in order to have a "Jugendtraum" for a number field $K$, you first want to have a complete class field theory for it.
Kronecker didn't have a general CFT, so his conjecture re …
- 17.6k
2
votes
Accepted
Counting number of primes that split completely in a number field
The unconditional counterpart to that estimate is
$$\pi_L(x)=\frac{\mathrm{Li}(x)}{[L:\mathbb{Q}]}+\frac{\mathrm{Li}(x^\beta)}{[L:\mathbb{Q}]}+c_1|\tilde{C}|x\exp(-c_2 n_L^{-1/2} \log^{1/2}x)$$
for …
- 17.6k
2
votes
Arakelov divisor on $\operatorname{Spec } O_F$: places or embeddings?
I think the natural definition is the one using archimedean places (or equivalently, all embeddings, with the pairs of complex conjugates considered as a single embedding), and I'm very curious of the …
- 17.6k
5
votes
Euclidean real quadratic fields
The problem of finding explicit Euclidean functions is even more complicated.
Usually the euclidean property is proved indirectly, using sieve-theoretic methods (in particular, Wilson's large sieve o …
- 17.6k
3
votes
Completion and algebraic closure
The completion of the algebraic closure of a valued field is algebraically closed and complete.
So any further operation of closure or completion gives you a field isomorphic to $\hat{\bar{K}}$.
- 17.6k