All Questions
6
questions
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votes
1
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69
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Reducible/Irreducible Polynomials in Ring Theory
I have this following exercise I've been trying to solve for a while now.
We are supposed to study the irreducibility of the polynomial $A=X^4 +1$ in $\mathbb{Z}[X]$ and in $\mathbb{Z}/p \mathbb{Z}$ ...
- 11
-2
votes
1
answer
54
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What do you call rings that have unique factorizations?
For example, integers, gaussian integers, and polynomials all have unique factorizations. What are these rings (or this property) referred to as? Or is unique factorization a ubiquitous property that ...
- 2,230
1
vote
0
answers
61
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minimality of ideals generated by primes among the prime ideals
I have following task:
Given a factorial ring $R$ (i.e. with unique prime decomposition) and a prime element $a \in R$ prove that if $I \subseteq R$ is a prime ideal with $(0) \subseteq I \subseteq ...
- 16.7k
0
votes
1
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110
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Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$? [closed]
The question is as in the title:
Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$?
- 157
2
votes
1
answer
484
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A non-UFD where prime=irreducible [duplicate]
It is easy to see that in an atomic domain (where every element factors into irreducibles), we have that all irreducibles are prime iff the domain in question is an UFD.
I think it is not true for a ...
- 36k
6
votes
1
answer
148
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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