All Questions
Tagged with nt.number-theory ac.commutative-algebra
360 questions
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Unique factorization in polynomial rings
Everybody knows that polynomial rings over fields have unique factorization, and that if $R$ has unique factorization, then so does $R[X]$. And everybody knows who proved these results first.
Well, ...
78
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9
answers
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Irreducibility of polynomials in two variables
Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper (Brindza and Pintér, On the irreducibility of some polynomials in ...
- 30.5k
14
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3
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non-Dedekind Domain in which every ideal is generated by at most two elements
Does anyone know of such a domain?
- 467
4
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1
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Do n-th Witt polynomials generate {P | P' is divisible by n} ?
EDIT: Proved it on my own. It easily follows from the Witt integrality theorem. Sorry for posting.
Let $P\in\mathbb{Z}\left[\Xi\right]$ be a polynomial (where $\Xi$ is a family of symbols that we use ...
- 33.8k
27
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5
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Class number measuring the failure of unique factorization
The statement that the class number measures the failure of the ring of integers to be a ufd is very common in books. ufd iff class number is 1. This inspires the following question:
Is there a ...
- 4,593
84
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31
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70k
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Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
6
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3
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Factorization of elements vs. of ideals, and is being a UFD equivalent to any property which can be stated entirely without reference to ring elements?
Why exactly is the unique factorization of elements into irreducibles a natural thing to look for? Of course, it's true in $\mathbb{Z}$ and we'd like to see where else it is true; also, regardless of ...
- 6,792
12
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3
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Dirichlet series with integer coefficients as a UFD
I recall the following question from Ulam's book "Unsolved math problems": show that the ring of Dirichlet series with integer coefficients is a factorial ring. I believe that soon after Ulam wrote ...
- 16.9k
16
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6
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Solving polynomial equations when you know in which number field the solutions live
Suppose I have a bunch of polynomial equations with coefficients in a number field, and suppose further that I'm guaranteed a priori that they have a solution in that number field. Can I leverage ...
- 28.1k
36
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4
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What is interesting/useful about big Witt Vectors?
$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing $\...
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