All Questions
Tagged with prime-factorization abstract-algebra
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
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
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
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
2
votes
1
answer
306
views
A simple algebraic ring extension of a UFD having no prime elements
Let $D$ be a UFD over a field $k$ of characteristic zero.
Assume that $w$ is algebraic over $D$.
Denote $R=D[w]$.
Observe that $R$ is not necessarily a UFD.
Can one find an example in which $R$ ...
- 6,705
2
votes
3
answers
236
views
Number of solutions of polynomials in a field
Consider the polynomial $x^2+x=0$ over $\mathbb Z/n\mathbb Z$
a)Find an n such that the equation has at least 4 solutions
b)Find an n such that the equation has at least 8 solutions
My idea is to ...
- 171
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
2
votes
2
answers
45
views
Does the monoid of non-zero representations with the tensor product admit unique factorization?
Let $(M, \cdot, 1)$ be a monoid. We will now define the notion of unique factorization monoid. A non-invertible element in $M$ is called irreducible if it cannot be written as the product of two other ...
- 1,649
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
0
votes
1
answer
175
views
Prove that the elements $2x$ and $x^2$ have no LCM in the ring of integral polynomials with even coefficient of $x$
Let $A$ be the subring of $\Bbb Z[x]$ consisting of all polynomials with even coefficient of $x$. Prove that the elements $2x$ and $x^2$ have no lowest common multiple.
Hints please!
- 1,332
0
votes
1
answer
186
views
Prime elements of $\mathbb{Z}[i\sqrt5]$.
I was studying the Gaussian integers and I proved that every composite number in $\mathbb{N}$ is not a prime in $\mathbb{Z}[i]$. This is true because this ring is an Euclidean domain, and if $n=ab$ is ...
user723846