All Questions
Tagged with prime-factorization abstract-algebra
20
questions with no upvoted or accepted answers
3
votes
0
answers
33
views
least upper bounds that are coprime
Given $n$ natural numbers $p_1$, $p_2$, ... $p_n$ find numbers $q_1$, $q_2$, ... $q_n$ that are pairwise coprimes such that $p_i$ ≤ $q_i$ and such that $\prod_{i=1..n} q_i$ is smallest possible.
I ...
- 31
3
votes
0
answers
60
views
What are the primes of $\mathbb{Z}[\sqrt{-p}\colon \text{$p$ prime}]$?
Let $R$ denote subdomain of $\mathbb{C}$ which is isomorphic to the direct limit of the diagram of commutative rings $\mathbb{Z}\hookrightarrow\mathbb{Z}[\sqrt{-2}]\hookrightarrow\mathbb{Z}[\sqrt{-2},\...
- 1,278
3
votes
0
answers
180
views
Does every commutative ring with $1 \neq 0$ and a non-zero, non-unit element contain a prime element?
As above, I am trying to answer the following question:
"Does every commutative ring with $1 \neq 0$ and a non-zero, non-unit element contain a prime element?"
Since in some books prime elements are ...
- 4,899
3
votes
0
answers
153
views
generalizing unique factorization domains by allowing *infinite* factorizations
When we consider an unique factorization domain, we get a factorization with finitely many factors. Is it possible to generalize an unique factorization domain by still requiring an unique ...
- 16.7k
2
votes
0
answers
92
views
Non-constant polynomial over an integral domain without any irreducible factors.
Let $R$ be an integral domain. I am trying to find a $f \in R[x]$, such that $\deg(f) \geq 1$, and $f$ does not have any irreducible factors in $R[x]$.
Does such $f$ exist?
Though I haven't been able ...
- 69
2
votes
0
answers
32
views
reducuble polynomial in two variables
Let $f(x,y)=x^4-y^3 \in \mathbb{R}[x,y]$. I claim that $f$ is reducible but i don`t know how to find its irrdeucible factors. Can anyone help me?
Thanks in advance.
- 1,071
2
votes
0
answers
59
views
Characterizing commutative semigroups with a factorization property.
Let $(N, \times)$ be a commutative semigroup and assume that a countably infinite subset $P$ of $N$ algebraically generates $N$, and let ${\mathcal F}(P)$ denote the set of all non-empty finite ...
- 11.5k
2
votes
0
answers
202
views
Non-unique factorization in $\mathbb{Z}[\sqrt{-5}]$
I want to show that the decomposition into irreducible factors in the ring
$$\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5}|\space a, b \in \mathbb{Z}\}$$
is not unique, except for the order of factors ...
- 3,057
2
votes
0
answers
56
views
If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$
If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$.
So $n^2-3 = pm$ for some integer $m$, and I know $|a^2 - ...
- 445
2
votes
0
answers
99
views
Lattice Bases of Prime (Ideal) Divisors
My question is:
How can I find the prime (ideal) divisors of 2 and 3 in the ring of integers of $\mathbb Q[\sqrt{-14}]$ and $\mathbb Q[\sqrt{-23}]$?
Here's what I have so far.
I found that (2, $\...
- 311
2
votes
0
answers
264
views
Unique factorization in 3-sphere coordinate ring
For $n\geqslant 1$ define $$A_n=\mathbb{C}[X_0,X_1,\dots,X_n]\Bigg/\left(\sum_{i=0}^{n}X^2_i-1\right).$$
I would like to prove that $A_3$ is a unique factorization domain.
For $A_2$ it is not true ...
- 913
1
vote
0
answers
67
views
Let $\mathbb{Z}[i]$ denote the *Gaussian integers*. Factor both $3+i$ and its norm into primes in $\mathbb{Z}[i]$
Question: Let $\mathbb{Z}[i]$ denote the Gaussian integers.
(a) Compute the norm $N(3+i)$ of $3+i$ in $\mathbb{Z}[i]$
(b) Factor both $3+i$ and its norm into primes in $\mathbb{Z}[i]$
(c) Compute $\...
- 1,349
1
vote
0
answers
117
views
Factorization in a principal ideal ring/rng
It is known that every PID is a UFD.
Is it true that every element of a commutative principal ideal ring (PIR) or rng that is not zero and not a unit is a product of a finite number of irreducible ...
- 1,120
1
vote
1
answer
177
views
Ring with infinitely reducible elements
Can you give or construct an elementary example of a factorial ring with elements which are product of infinitely many irreducible elements? i.e. there are reducible elements that can't be written as ...
- 192
1
vote
0
answers
61
views
minimality of ideals generated by primes among the prime ideals
I have following task:
Given a factorial ring $R$ (i.e. with unique prime decomposition) and a prime element $a \in R$ prove that if $I \subseteq R$ is a prime ideal with $(0) \subseteq I \subseteq ...
- 16.7k