A commutative ring with unity in which every nonzero, nonunit element can be written as a product of irreducible elements, and where such product is unique up to ordering and associates. Also called UFD or "factorial ring"
Questions tagged [unique-factorization-domains]
687 questions
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How to determine this solutions?
(a) Determine all positive rational solutions of $x^y=y^x$
(b) Determine all positive rational solutions of $x^{x+y}=(x+y)^y$
K.C.S.
- 183
2
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1 answer
Why is the quadratic field $\Bbb Q [\sqrt{-7}]$ a unique factorization domain?
I can not find an answer to this that makes any sense to me. I am hoping someone can dumb it down a little or show me something I am missing.
But I can not seem to understand why the Quadratic Field of
$\Bbb Q [\sqrt{-7}]$ has unique…
Seacomit
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2 answers
is there any subset of the reals that is not a Unique Factorization Domain?
Is there any subset of the real numbers that is not a Unique Factorization Domain? (i.e. where within that subset, a "prime" is a number that cannot be written as a product of any numbers in that set except itself and 1, and where there is at least…
Bennett
- 203
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1 answer
List of quadratic field with the UFD property
Let D be a square free integer let $K=\mathbb{Q}(\sqrt{D})$ and let
$\mathcal{O_{K}}$ be the ring of integer of $K$
My question: where I can find a list of of the value of $D$ which makes the ring of the quadratic field
$K=\mathbb{Q}(\sqrt{D})$…
Aster Phoenix
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If $p$ is prime and $d$ is a square mod $p$, then prove that $p$ is reducible in $\mathbb{Z}[\sqrt{d}]$.
Suppose $d$ be a square-free integer and $\mathbb{Z}[\sqrt{d}]$ is a unique factorization domain.
If $p$ is prime and $d$ is a square mod $p$, then prove that $p$ is reducible in $\mathbb{Z}[\sqrt{d}]$.
Assume that $p$ is prime and $d$ is a square…
xxxxxx
- 603
0
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2 answers
Unique Factorization Up To Units
So a lot of people are like $\mathbb{Z}$ has a unique factorization up to units...but that doesn't make sense to me because (for example), $(-1) \cdot (-1) \cdot 2 \cdot 3$ should be a factorization; but then $2 \cdot 3$ is also a factorization, so…
Wa Wa
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1 answer
A confusion about UFDs
So I'm new to this concept and when I was reading the definition.
It says that if R is a UFD then every non zero element can be written as the product of irreducible elements.
So here's my question which is stupid I know.
Are these irreducible…
Hassuni
- 143
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GCD in unique factorization domain
Let be D a unique factorization domain, k in D and $x=kd $ with $d$ a gcd of $a, b$ and $y$ a gcd of $ka, kb$ prove that x and y are associates; x divides y and y divides x.
My attempt:
x=kd and a=dr, b=ds hence ka=kdr, kb=kds so ka=xr, kb=xs hence…
asv
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In a UFD, why does $ab | c^n$ implies $ab | c$?
In a UFD, if I have two irreducible elemnts, $a$ and $b$ such that $ab | c^n$, why does this imply that $ab | c$?
edit
Here $a$ and $b$ are different and coprime.
roi_saumon
- 4,196
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Generate guaranteed unique number from given inputs that is only 7 digits
The short version: How can I use minutes since a recent date and 2 sets of known numbers from 0-255 to guarantee a unique 7 digit number?
Long version:
I'm trying to generate a unique number or string that is guaranteed to be unique for at least 5…
luhlig
- 1
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1 answer
Is $F_p^{l}[t]$ is a UFD
I'm stuck here.
Let $F_{p}$ be a finite filed. Is $F_{p^{l}}[t]$ a Unique factorization Domain.
Can anyone explain? Thank you very much
Math123
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unit element in unique factorization domain
Integral domain is unique factorization domain if every non zero , non unit element can be written uniquely as finite product of irreducible elements .
why in this definition non unit element is required?
what happens if i will take unit element ?…
dipali mali
- 555
0
votes
1 answer
Value of $a$ for $x^{2}-4x+4-2a=0$ to have a unique solution
By using the root formula i got $(2\pm \sqrt{8a})$ expression.
So, $ a $ cannot be negative and if $a$ is positive we get two solutions.
Therefore, $a=0$ is the answer??
Is my approach correct and is there a better way to solve this problem?
user405925
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A paper on the relation between Ernst Kummer's work on factorization and structural linguistics (Zellig Harris)
Well, since I have been consulting this forum pertaining to the topic of this thread, I find it fair to provide you with the link to my newly-published paper in
http://revistadefilosofia.com/64-03.pdf (one of the most relevant philosophical journals…
Javier Arias
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