All Questions
9 questions
17
votes
1
answer
754
views
Principal ideal domains with finitely many units
Question: What are the (in characteristic 0 if needed) principal ideal domains that have finitely many units?
Can such rings be classified?
(This is a more specialised version of the question in ...
- 26.5k
2
votes
0
answers
491
views
Examples of almost Dedekind domains that are not Dedekind
All I know about almost Dedekind domains (which I have come to learn about only recently) is that they are integral domains whose localization at every prime is a discrete valuation ring. In other ...
- 739
5
votes
1
answer
959
views
Rings of $S$-integers are finitely generated as rings
Let $K$ be a global field (number field or algebraic function field over a finite field), $\mathcal{V}$ the set of $\mathbb{Z}$-valuations on $K$, $S \subseteq \mathcal{V}$ a finite set. The ring of $...
- 277
1
vote
1
answer
142
views
Valuation theory on semisimple algebras used in the paper of Cohen-Martinet: reference request
I'm currently reading the paper of Henri Cohen & Jacques Martinet "Etude heuristique des groupes de classes des corps de nombres"
On the 2nd section, they recall some facts on valuations, ...
- 1,013
1
vote
1
answer
256
views
Lattice Sieving
What are some good references for Lattice Sieving in Number Field Sieve? Could someone suggest some research papers in this area?(Theoretical and Computational Perspective)
- 63
5
votes
1
answer
2k
views
Generalizing Dedekind's Factorization Theorem
A classical theorem due to Dedekind states the following:
Let $O_{K}$ be the ring of integers of a number field $K$, and assume
$K$ is generated by adjoining the algebraic integer $\alpha$ to
$...
- 14.6k
24
votes
3
answers
4k
views
Commutative algebra with a view toward algebraic _number theory_
Someone asked me this today, and I don't know what the standard answer is:
Is there an analogue of David Eisenbud's rather amazing Commutative Algebra With a View Toward Algebraic Geometry but with a ...
6
votes
2
answers
850
views
Decomposition of finite algebras over finite fields
Let $K$ be a number field, $Z_K$ its ring of integers, and $p$ a rational prime number. Then $A_p = Z_K/(p)$ is a finite ${\mathbb F}_p$-algebra. Using ideal arithmetic in $Z_K$ and the Chinese ...
- 32.6k
6
votes
1
answer
641
views
The Jacobian ideal generates the socle of a complete intersection
This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here:
http://tinyurl.com/2967eov
I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ ...
- 805