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Results tagged with algebraic-number-theory
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user 32332
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
9
votes
Questions about the proof of Stickelberger's theorem on discriminants
A bit too long for a comment, so I write it as an answer.
$(1)$ Let $L$ be the Galois closure of $K$ over $\mathbb{Q}$.
We can apply $\sigma\in Gal(L,\mathbb{Q})$ on $P+N$ and $PN$, which lie in $L$. …
- 12.1k
2
votes
Results for resolution of equations in polynomial ring
In general this is a difficult problem, but in special cases like $x^n+y^n=z^n$ in polynomials we can use Mason's theorem, which is about an analogue of the $abc$-conjecture for polynomials in $\mathb …
- 12.1k
9
votes
0
answers
605
views
Does Hasse-Minkowski help to produce nontrivial rational solutions?
Consider a quadratic form over $\mathbb{Q}$, say, a diagonal one in three variables
$$
F(X, Y,Z) = a · X^2 + b · Y^2 − c · Z^2
$$
with positive integers $a,b,c$. Then $F(X,Y,Z)=0$ has a non-trivial ra …
- 12.1k
10
votes
Question about ring of integers of cyclotomic field
The ring of integers $\mathbb{Z}[\zeta_p]$ is an UFD if and only if the class number of $\mathbb{Q}[\zeta_p]$ is $1$. This is the case if and only if $p\le 19$. For bigger primes $p$ the class numbers …
- 12.1k
6
votes
Cohen-Lenstra Heuristics reference
There is among other references the thesis of J. Lengler devoted to the Cohen-lenstra heuristics. This is not too technical and has many details. The author says: "
The aim of this thesis is to explai …
- 12.1k
3
votes
Text for Algebraic Number Theory
I think the book Algebraic number theory by Helmut Koch should be mentioned too, together with his book Number Theory: Algebraic Numbers and Functions. For an overview and a discussion see the talk gi …