All Questions
Tagged with prime-factorization abstract-algebra
97
questions
35
votes
3
answers
17k
views
What are examples of irreducible but not prime elements?
I am looking for a ring element which is irreducible but not prime.
So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\in R$.
This is irreducible because in any ...
- 1,057
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
13
votes
2
answers
3k
views
$K[x_1, x_2,\dots ]$ is a UFD
I wonder about how to conclude that $R=K[x_1, x_2,\dots ]$ is a UFD for $K$ a field.
If $f\in R$ then $f$ is a polynomial in only finitely many variables, how do I prove that any factorization of $f$...
- 2,157
8
votes
5
answers
14k
views
Proving gcd($a,b$)lcm($a,b$) = $|ab|$
Let $a$ and $b$ be two integers. Prove that $$ dm = \left|ab\right| ,$$ where $d = \gcd\left(a,b\right)$ and $m = \operatorname{lcm}\left(a,b\right)$.
So I went about by saying that $a = p_1p_2......
- 2,148
7
votes
2
answers
4k
views
Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$
$d \in \mathbb{Z}$ is a square-free integer ($d \ne 1$, and $d$ has no factors of the form $c^2$ except $c = \pm 1$), and let $R=\mathbb{Z}[\sqrt{d}]= \{ a+b\sqrt{d} \mid a,b \in \mathbb{Z} \}$. Prove ...
- 3,269
7
votes
1
answer
211
views
Irreducibility of an integer polynomial with exponents in linear sequence?
Let $b$ and $n$ be two positive integers. Is there are a general result which tell us when the polynomial $$1+x^{b}+x^{2b}+x^{3b}+\cdots+x^{nb}$$
is irreducible over the integers?
I know that $$1+x+...
user123641
6
votes
1
answer
148
views
factoring $x^n+x+1$
Is there a way of factoring a polynomial of the general form $$x^n+x+1$$ in the ring $\mathbb C[x]$ or $\mathbb R[x]$ or $\mathbb Z [x]$ for any $n \in \mathbb N$? (Or perhaps with certain conditions ...
- 16.7k
5
votes
3
answers
183
views
Sanity check on factorization of $\langle 5 \rangle$ in $\mathbb{Z}_{10}$
Looking at $\mathbb{Z}_{10}$, consisting of $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ with addition and multiplication adjusted so the ring is closed under both operations.
Clearly $5 = 3 \times 5 = 5 \times 5 =...
- 1,108
4
votes
1
answer
2k
views
How can I find decompositions in $\mathbb{Z}[\sqrt{d}]$?
Decompositions in $\mathbb{Z}$
In $\mathbb{Z}$ you can find a decomposition of any element $n \in \mathbb{Z}$ by factorization such that
$$n = \prod_{p \in \mathbb{P}} p^{v_p(n)}$$
So for a ...
- 9,891
4
votes
2
answers
1k
views
Verifying prime factorization equivalence class
I define a relation on $\Bbb N$ as follows:
$x \sim y \Longleftrightarrow \ \exists \ j,k \in \Bbb Z$ s.t. $x \mid y^j \ \wedge \ y \mid x^k$
I have shown that $\sim$ is an equivalence relation by ...
- 4,467
3
votes
2
answers
682
views
Definition of UFD and the fact that UFDs are integrally closed
I am trying to understand the proof of the fact that UFDs are integrally closed. In addition to the lecture notes I have, there are at least two solutions on MSE:
One is here: How to prove that UFD ...
- 603
3
votes
1
answer
84
views
A non-UFD $B$ such that $A \subset B \subset C$, where $A \cong C$ are UFD's
Let $A \subset C$ be two isomorphic unique factorization domains (UFD's).
Is it possible to find an integral domain $B$, $A \subset B \subset C$,
such that $B$ is not a UFD?
I have tried to ...
- 6,705
3
votes
3
answers
764
views
Is $\mathbb{Z}[√13] $ a Unique factorization domain?
I think it is not so as $12 $ can be written in two ways $12=2.6=(1+\sqrt{13})(-1+\sqrt{13})$. Are these two factorization unique upto irreducibles? Please help.
- 241
3
votes
1
answer
480
views
Why a certain integral domain is not a UFD.
Let
$$\mathbb{Z}[q]^{\mathbb{N}} = \varprojlim_j \mathbb{Z}[q]/((1-q)\cdots (1- q^j))$$
Why isn't $\mathbb{Z}[q]^{\mathbb{N}}$ a unique factorization domain?
The author proposes a proof whose ...
- 2,236
3
votes
1
answer
688
views
Primary Decomposition
The standard primary decomposition theorem in algebra is about being able to write an ideal uniquely as an intersection of primary ideals. In linear algebra the theorem is about how a vector space can ...
- 1,457