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Results tagged with algebraic-number-theory
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user 16537
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
4
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1
answer
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A slick proof of "The ring of integers of a number field has infinitely many non-associated ...
Let $\mathbf Z_K$ be the ring of integers of an algebraic number field $K$. It is well known that $\mathbf Z_K$ has infinitely many non-associated atoms (and hence is not a Cohen-Kaplansky domain).
…
- 10.5k
4
votes
0
answers
67
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Counting incongruent isometric factorizations in the ring of integers of a number field with...
Let $H$ be a multiplicatively written commutative monoid. We use $\mathcal A(H)$ for the set of atoms of $H$ and $\pi_H$ for the canonical homomorphism $\mathscr F(\mathcal A(H)) \to H$, where $a \in …
- 10.5k
18
votes
5
answers
3k
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An elementary, short proof that the group of units of the ring of integers of a number field...
Dirichlet's unit theorem states that (i) the group of units, $\mathscr{U}_K$, of the ring of integers of a number field $K$ is finitely generated, and (ii) the rank of $\mathscr{U}_K$ is equal to $r_1 …
- 10.5k
5
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Dihedral extensions and the Ankeny–Artin–Chowla conjecture
It appears that my former officemate Andreas Reinhart (University of Graz, Austria) has disproven the Ankeny–Artin–Chowla conjecture: more precisely, Andreas has found that
$$
d := 331914313984493$$
i …
- 10.5k
1
vote
0
answers
50
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Closedness of the range of the distorsion of the multiplicative monoid of a number field
Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is an element $x \in H \setminus H^\times$ such that $a \ne xy$ for all $x, y \in H \setminus H^\times$, where $H^\tim …
- 10.5k
4
votes
1
answer
173
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On the factorization of powers of atoms in the ring of integers of a number field
Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is a non-unit element $a \in H$ that doesn't split into the product of two non-unit elements.
Given $x \in H$, we tak …
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