All Questions
Tagged with prime-factorization abstract-algebra
97
questions
0
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33
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A question about Gaussian integers [duplicate]
I am working on the following exercise:
Show that if $p \equiv 1 \mod 4$ then $p$ is not prime in $\mathbb{Z}[i]$, but instead splits as the product of two distinct prime. [Hint: Show that $p|(a^2+1)$ ...
- 1,065
1
vote
1
answer
314
views
Clarify about prime decomposition of a radical ideal
My teacher stated the following result:
Let $R$ be a ring, and consider $I=\sqrt I\subseteq A$ a radical ideal. $I$ is equal to the intersection of the primes minimal over $I$, i.e. the minimal (with ...
- 2,523
3
votes
0
answers
33
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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
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
2
votes
0
answers
92
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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
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
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66
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Why $r \notin R^*$ if it is a prime? [duplicate]
My professor gave us the following definition:
Let $R$ be a commutative ring. $r \in R$ is prime if $r.R$ is a prime ideal. Then he concluded that this definition tells us $r \notin R^*$ but I do not ...
user778657
0
votes
1
answer
501
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Polynomial in $\mathbb{Z}[x]$ which is irreducible but not prime. [duplicate]
I know that in an integral domain, prime implies irreducible. Moreover, in a principal integral domain, these notions are equivalent. The ring of polynomials $\mathbb{Z}[x]$ is not a principal ...
- 1,750
0
votes
1
answer
186
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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
0
votes
1
answer
94
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Disproving unique factorization of $\mathbb{Q}(\sqrt{-6})$
For homework, we were given a problem that asked to explain why $(\sqrt{-6})(-\sqrt{-6})$ implies $\mathbb{Q}(\sqrt{-6})$ does not have a unique factorization.
I understand the unique part in this ...
- 9
0
votes
1
answer
81
views
Why does $(5+\sqrt{3})(5-\sqrt{3})$ not conflict with $\mathbb{Q}(\sqrt{3})$ having unique factorization?
My intuition is to try to show that there is some irreducible element $p \in \mathbb{Q}(5)$ that divides $(5+\sqrt{3})(5-\sqrt{3})$, but I'm having trouble finding it. Is there an easier way to do ...
- 1
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
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
2
votes
2
answers
267
views
Prime element in $\mathbb{Z}[\sqrt{p}]$ [duplicate]
For which prime integers $p$ is $5$ a prime in $\mathbb{Z}[\sqrt{p}]$, the subring of the reals generated by the integers and $\sqrt{p}$ ?
I know that $\mathbb{Z}[\sqrt{p}] = \{a + b\sqrt{p} | a,b \...
- 1,503
2
votes
0
answers
32
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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