All Questions
Tagged with prime-factorization unique-factorization-domains
34
questions
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2
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607
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Factorization of Gauss integers
How do I find the factorization in prime elements of $20538 - 110334i$ in $\Bbb{Z}[i]$? I have found that $20538=2×3^2×7×163$ and $110334=2×3×7×2627$ but I don't know how to use this.
user372244
1
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2
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136
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Justifying the representation of a monic polynomial over a UFD
Let $D$ be a UFD and $f(x) \in D[x]$ be monic. The book I'm reading from claims that $$f(x) = p_1(x)^{e_1} \cdots p_n(x)^{e_n}$$
where $p_i(x)$ are distinct, irreducible, and monic, and $e_i >0$.
...
- 4,039
2
votes
1
answer
306
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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
3
votes
1
answer
84
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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
2
votes
1
answer
154
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Proof help involving prime factorization, exponents
Let $a,b\in\mathbb{N}$. Moreover, let $p_1,p_2,...,p_k$ be the collection of all primes which divide $a$ or $b$ or both. We'll write $a=p_1^{\alpha_1}p_2^{\alpha_2}...p_k^{\alpha_k}$ and $b=p_1^{\...
user322548
1
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1
answer
172
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Reducing Godel number to a manageable size
I have a set of $n$ attributes $A$, i.e. {$A_1, A_2, .. , A_n$}. Each attribute has $m : m > 0$ possible values, which we can call $V_{a,m}$.
I also have a population of objects, which are ...
- 144
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153
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generalizing unique factorization domains by allowing *infinite* factorizations
When we consider an unique factorization domain, we get a factorization with finitely many factors. Is it possible to generalize an unique factorization domain by still requiring an unique ...
- 16.7k
1
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0
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51
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Properties involving prime factorization and divisibility
Can anyone help me out this with proof?
Let n be a positive integer greater than 1 with the property that whenever n divides a product ab where a,b ∈ Z, then n divides a or n divides b. Prove that ...
- 73
3
votes
1
answer
49
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Prime and Unique Factorization Proof
I could use some help with this question here:
Let $n ∈ Z, n > 1$. Prove that if n is not divisible by any prime number less than or equal to $√n$, then n is a prime number.
Here I assumed that ...
- 73
2
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0
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142
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Notation for indexing the factorizations of a number?
Background
Given any $n \in \mathbb{N}$, the ordered factorization count of $n$ can be computed and is traditionally written $H(n)$. This is, essentially, the number of unique decompositions of $n$ ...
- 341
4
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2
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1k
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How does one prove that $\mathbb{Z}\left[\frac{1 + \sqrt{-43}}{2}\right]$ is a unique factorization domain?
By extending the Euclidean algorithm one can show that $\mathbb{Z}[i]$ has unique factorization.
This logic extends to show $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{2}\right]$, $\mathbb{Z}\left[\frac{1 +...
- 24.5k
3
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1
answer
2k
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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
1
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2
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438
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Where does the proof of unique factorization fail for $\mathbb Z[\sqrt{-5}]$?
I know that unique factorization does not hold for all rings, such has the much-used example $\mathbb Z[\sqrt{-5}]$. It seems that Euclid's lemma does not hold for these rings, and so on. However, ...
- 1,790
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73
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If $a^2+3b^2$ is a cube in $\mathbb Z$ , then are $a+\sqrt{-3}b$ and $a-\sqrt{-3}b$ both cubes in $\mathbb Z[\sqrt{-3}]$ ?
If $a,b \in \mathbb Z$ are such that g.c.d.$(a,b)=1$ and if $a^2+3b^2$ is a cube in $\mathbb Z$ , then are $a+\sqrt{-3}b$ and $a-\sqrt{-3}b$ both cubes in $\mathbb Z[\sqrt{-3}]$ ? I cannot use ...
user217921
2
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1
answer
162
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Can you give an example of an irreducible element of the ring of Dirichlet series with integer coefficients?
According to this. The ring of Dirichlet series with integer coefficients is a UFD. Can you give an example of an irreducible element in that ring?
