All Questions
13
questions
3
votes
2
answers
682
views
Definition of UFD and the fact that UFDs are integrally closed
I am trying to understand the proof of the fact that UFDs are integrally closed. In addition to the lecture notes I have, there are at least two solutions on MSE:
One is here: How to prove that UFD ...
- 603
0
votes
1
answer
652
views
Prime elements of the ring $\mathbb Z[i]$.
I want to find the prime elements of the ring $\mathbb Z[i]$ of Gaussian integers.Define $d(a+ib)=a^2+b^2$ for all $a+ib\in \mathbb Z[i]-\{0\}$.I think if $d(a+ib)$ is prime in $\mathbb Z$ then and ...
- 3,619
1
vote
2
answers
365
views
If $R$ is a UFD, then $R[x]$ is Noetherian?
If $R$ is Noetherian, then $R[x]$ is Noetherian. However, if $R$ is a UFD, then $R$ might not be Noetherian. $\DeclareMathOperator{\Frac}{Frac}$
I want to show that if $R$ is a UFD, then $R[x]$ is a ...
- 1,751
3
votes
3
answers
764
views
Is $\mathbb{Z}[√13] $ a Unique factorization domain?
I think it is not so as $12 $ can be written in two ways $12=2.6=(1+\sqrt{13})(-1+\sqrt{13})$. Are these two factorization unique upto irreducibles? Please help.
- 241
0
votes
2
answers
138
views
Number of irreducible divisors in a UFD
The following fact is totally obvious, but I cannot find a way to prove it.
Let $R$ be a UFD and $a \in R$ be non zero and non invertible. Factor it as a product of irreducible elements: $a=p_1 \cdot ...
- 579
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
3
votes
1
answer
386
views
Number of prime elements in a ring
Is there any way to count the prime elements in a ring?
More precisely a way to count prime elements in a UFD? Which would be the same as counting irreducibles. Are there even UFD with finitely many ...
- 547
17
votes
1
answer
963
views
What is the correct notion of unique factorization in a ring?
I was recently writing some notes on basic commutative ring theory, and was trying to convince myself why it was a good idea to study integral domains when it comes to unique factorization.
If $R$ is ...
- 997
0
votes
2
answers
607
views
Factorization of Gauss integers
How do I find the factorization in prime elements of $20538 - 110334i$ in $\Bbb{Z}[i]$? I have found that $20538=2×3^2×7×163$ and $110334=2×3×7×2627$ but I don't know how to use this.
user372244
4
votes
2
answers
1k
views
How does one prove that $\mathbb{Z}\left[\frac{1 + \sqrt{-43}}{2}\right]$ is a unique factorization domain?
By extending the Euclidean algorithm one can show that $\mathbb{Z}[i]$ has unique factorization.
This logic extends to show $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{2}\right]$, $\mathbb{Z}\left[\frac{1 +...
- 24.5k
3
votes
1
answer
2k
views
About the ways prove that a ring is a UFD.
I'm doing an exercise where I have a commutative ring with unity $R$. We had to find that the nonunits formed an ideal (maximal). After that, we found the irreducible elements, and then we saw that ...
- 4,225
1
vote
2
answers
438
views
Where does the proof of unique factorization fail for $\mathbb Z[\sqrt{-5}]$?
I know that unique factorization does not hold for all rings, such has the much-used example $\mathbb Z[\sqrt{-5}]$. It seems that Euclid's lemma does not hold for these rings, and so on. However, ...
- 1,790
1
vote
1
answer
73
views
If $a^2+3b^2$ is a cube in $\mathbb Z$ , then are $a+\sqrt{-3}b$ and $a-\sqrt{-3}b$ both cubes in $\mathbb Z[\sqrt{-3}]$ ?
If $a,b \in \mathbb Z$ are such that g.c.d.$(a,b)=1$ and if $a^2+3b^2$ is a cube in $\mathbb Z$ , then are $a+\sqrt{-3}b$ and $a-\sqrt{-3}b$ both cubes in $\mathbb Z[\sqrt{-3}]$ ? I cannot use ...
user217921