All Questions
Tagged with prime-factorization abstract-algebra
97
questions
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
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
1
vote
2
answers
2k
views
Prove: There are zero divisors in $\Bbb Z_n$ if and only if $n$ is not prime.
I need to prove there are zero divisors in $\mathbb{Z}_n$ if and only if $n$ is not prime.
What should I consider first?
- 497
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
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
1
answer
176
views
If every irreducible element in $D$ is prime, then $D$ has the unique factorization property.
Suppose every irreducible element in a domain $D$ is prime.
I'm trying to prove this implication:
In a integral domain $D$, if $a=c_1c_2...c_n$ and $a=d_1d_2...d_m$
($c_i,d_i$ irreducible), then ...
- 603
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