Zermelo's proof for unique factorisationUnique factorization in polynomial ringsAlternative proof of unique factorization for ideals in a Dedekind ringUnique factorisation and the fact that $mathbb A^2-0$ is not an affine variety?Well founded induction attributed to NoetherFactorisation of a biquadratic polynomialSandwich theorem for UFD'sUnique factorisation of prime geodesics?
Zermelo's proof for unique factorisation
Unique factorization in polynomial ringsAlternative proof of unique factorization for ideals in a Dedekind ringUnique factorisation and the fact that $mathbb A^2-0$ is not an affine variety?Well founded induction attributed to NoetherFactorisation of a biquadratic polynomialSandwich theorem for UFD'sUnique factorisation of prime geodesics?
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In Peter Bundschuh's "Einführung in die Zahlentheorie" I came across a possibly well-known but to me rather peculiar proof of unique factorisation, which is attributed to Ernst Zermelo. The proof bypasses Euclid's lemma to prove that $ mathbbZ$ is a UFD. It seems to me that the deepest property of the ring of integers used is some properties of the order on $ mathbbZ$! An account of this proof can be seen here:
https://planetmath.org/inductionproofoffundamentaltheoremofarithmetic
Can this idea be applied or has it been applied to any other situations? Or is it simply a tricky proof for the ring of integers?
nt.number-theory ac.commutative-algebra unique-factorization-domains
edited 8 hours ago
Matt F.
8,3071 gold badge20 silver badges52 bronze badges
asked 8 hours ago
Keivan KaraiKeivan Karai
3,5871 gold badge21 silver badges39 bronze badges
$endgroup$
add a comment |
$begingroup$
In Peter Bundschuh's "Einführung in die Zahlentheorie" I came across a possibly well-known but to me rather peculiar proof of unique factorisation, which is attributed to Ernst Zermelo. The proof bypasses Euclid's lemma to prove that $ mathbbZ$ is a UFD. It seems to me that the deepest property of the ring of integers used is some properties of the order on $ mathbbZ$! An account of this proof can be seen here:
https://planetmath.org/inductionproofoffundamentaltheoremofarithmetic
Can this idea be applied or has it been applied to any other situations? Or is it simply a tricky proof for the ring of integers?
nt.number-theory ac.commutative-algebra unique-factorization-domains
edited 8 hours ago
Matt F.
8,3071 gold badge20 silver badges52 bronze badges
asked 8 hours ago
Keivan KaraiKeivan Karai
3,5871 gold badge21 silver badges39 bronze badges
$endgroup$
1
$begingroup$
Second-last line: "'the prime p is in the prime decomposition of (q-p)c and thus also at least of q-p or c. But we know that p∤c, whence p∣q-p." Are you using Euclid Lemma here, or am I missing something?
$endgroup$
– Francesco Polizzi
8 hours ago
$begingroup$
Since $n_0 < n$, by induction hypothesis $p$ must appear in the unique decomposition of $n_0 = (q-p)c$, which is the product of the unique decompositions of $q-p$ and $c$. It's true that the sentence is quite confusing.
$endgroup$
– Wille Liou
5 hours ago
add a comment |
$begingroup$
In Peter Bundschuh's "Einführung in die Zahlentheorie" I came across a possibly well-known but to me rather peculiar proof of unique factorisation, which is attributed to Ernst Zermelo. The proof bypasses Euclid's lemma to prove that $ mathbbZ$ is a UFD. It seems to me that the deepest property of the ring of integers used is some properties of the order on $ mathbbZ$! An account of this proof can be seen here:
https://planetmath.org/inductionproofoffundamentaltheoremofarithmetic
Can this idea be applied or has it been applied to any other situations? Or is it simply a tricky proof for the ring of integers?
nt.number-theory ac.commutative-algebra unique-factorization-domains
edited 8 hours ago
Matt F.
8,3071 gold badge20 silver badges52 bronze badges
asked 8 hours ago
Keivan KaraiKeivan Karai
3,5871 gold badge21 silver badges39 bronze badges
$endgroup$
In Peter Bundschuh's "Einführung in die Zahlentheorie" I came across a possibly well-known but to me rather peculiar proof of unique factorisation, which is attributed to Ernst Zermelo. The proof bypasses Euclid's lemma to prove that $ mathbbZ$ is a UFD. It seems to me that the deepest property of the ring of integers used is some properties of the order on $ mathbbZ$! An account of this proof can be seen here:
https://planetmath.org/inductionproofoffundamentaltheoremofarithmetic
Can this idea be applied or has it been applied to any other situations? Or is it simply a tricky proof for the ring of integers?
nt.number-theory ac.commutative-algebra unique-factorization-domains
nt.number-theory ac.commutative-algebra unique-factorization-domains
edited 8 hours ago
Matt F.
8,3071 gold badge20 silver badges52 bronze badges
asked 8 hours ago
Keivan KaraiKeivan Karai
3,5871 gold badge21 silver badges39 bronze badges
edited 8 hours ago
Matt F.
8,3071 gold badge20 silver badges52 bronze badges
asked 8 hours ago
Keivan KaraiKeivan Karai
3,5871 gold badge21 silver badges39 bronze badges
edited 8 hours ago
Matt F.
8,3071 gold badge20 silver badges52 bronze badges
edited 8 hours ago
Matt F.
8,3071 gold badge20 silver badges52 bronze badges
edited 8 hours ago
Matt F.
8,3071 gold badge20 silver badges52 bronze badges
8,3071 gold badge20 silver badges52 bronze badges
asked 8 hours ago
Keivan KaraiKeivan Karai
3,5871 gold badge21 silver badges39 bronze badges
asked 8 hours ago
Keivan KaraiKeivan Karai
3,5871 gold badge21 silver badges39 bronze badges
asked 8 hours ago
Keivan KaraiKeivan Karai
3,5871 gold badge21 silver badges39 bronze badges
3,5871 gold badge21 silver badges39 bronze badges
1
$begingroup$
Second-last line: "'the prime p is in the prime decomposition of (q-p)c and thus also at least of q-p or c. But we know that p∤c, whence p∣q-p." Are you using Euclid Lemma here, or am I missing something?
$endgroup$
– Francesco Polizzi
8 hours ago
$begingroup$
Since $n_0 < n$, by induction hypothesis $p$ must appear in the unique decomposition of $n_0 = (q-p)c$, which is the product of the unique decompositions of $q-p$ and $c$. It's true that the sentence is quite confusing.
$endgroup$
– Wille Liou
5 hours ago
add a comment |
1
$begingroup$
Second-last line: "'the prime p is in the prime decomposition of (q-p)c and thus also at least of q-p or c. But we know that p∤c, whence p∣q-p." Are you using Euclid Lemma here, or am I missing something?
$endgroup$
– Francesco Polizzi
8 hours ago
$begingroup$
Since $n_0 < n$, by induction hypothesis $p$ must appear in the unique decomposition of $n_0 = (q-p)c$, which is the product of the unique decompositions of $q-p$ and $c$. It's true that the sentence is quite confusing.
$endgroup$
– Wille Liou
5 hours ago
1
1
$begingroup$
Second-last line: "'the prime p is in the prime decomposition of (q-p)c and thus also at least of q-p or c. But we know that p∤c, whence p∣q-p." Are you using Euclid Lemma here, or am I missing something?
$endgroup$
– Francesco Polizzi
8 hours ago
$begingroup$
Second-last line: "'the prime p is in the prime decomposition of (q-p)c and thus also at least of q-p or c. But we know that p∤c, whence p∣q-p." Are you using Euclid Lemma here, or am I missing something?
$endgroup$
– Francesco Polizzi
8 hours ago
$begingroup$
Since $n_0 < n$, by induction hypothesis $p$ must appear in the unique decomposition of $n_0 = (q-p)c$, which is the product of the unique decompositions of $q-p$ and $c$. It's true that the sentence is quite confusing.
$endgroup$
– Wille Liou
5 hours ago
$begingroup$
Since $n_0 < n$, by induction hypothesis $p$ must appear in the unique decomposition of $n_0 = (q-p)c$, which is the product of the unique decompositions of $q-p$ and $c$. It's true that the sentence is quite confusing.
$endgroup$
– Wille Liou
5 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Note that the popular textbook by Niven, Zuckerman, and Montgomery also has a proof of unique factorization in $mathbbZ$ that is based only on the well-ordering property and that bypasses Euclid's lemma.
The ascending chain condition in the definition of Dedekind domains is used in a similar way to the well-ordering principle in proving unique factorization of ideals in such domains.
answered 8 hours ago
MichaelGaudreauMichaelGaudreau
2842 silver badges6 bronze badges
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add a comment |
$begingroup$
In Pete Clark's notes on factorization, he calls this the "Lindemann–Zermelo proof" of the fundamental theorem of arithmetic, and he uses a similar idea to prove the well-known fact that if $R$ is a unique factorization domain, then so is $R[t]$. See Theorem 27.
answered 1 hour ago
Timothy ChowTimothy Chow
37k14 gold badges189 silver badges334 bronze badges
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2 Answers
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2 Answers
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$begingroup$
Note that the popular textbook by Niven, Zuckerman, and Montgomery also has a proof of unique factorization in $mathbbZ$ that is based only on the well-ordering property and that bypasses Euclid's lemma.
The ascending chain condition in the definition of Dedekind domains is used in a similar way to the well-ordering principle in proving unique factorization of ideals in such domains.
answered 8 hours ago
MichaelGaudreauMichaelGaudreau
2842 silver badges6 bronze badges
$endgroup$
add a comment |
$begingroup$
Note that the popular textbook by Niven, Zuckerman, and Montgomery also has a proof of unique factorization in $mathbbZ$ that is based only on the well-ordering property and that bypasses Euclid's lemma.
The ascending chain condition in the definition of Dedekind domains is used in a similar way to the well-ordering principle in proving unique factorization of ideals in such domains.
answered 8 hours ago
MichaelGaudreauMichaelGaudreau
2842 silver badges6 bronze badges
$endgroup$
add a comment |
$begingroup$
Note that the popular textbook by Niven, Zuckerman, and Montgomery also has a proof of unique factorization in $mathbbZ$ that is based only on the well-ordering property and that bypasses Euclid's lemma.
The ascending chain condition in the definition of Dedekind domains is used in a similar way to the well-ordering principle in proving unique factorization of ideals in such domains.
answered 8 hours ago
MichaelGaudreauMichaelGaudreau
2842 silver badges6 bronze badges
$endgroup$
Note that the popular textbook by Niven, Zuckerman, and Montgomery also has a proof of unique factorization in $mathbbZ$ that is based only on the well-ordering property and that bypasses Euclid's lemma.
The ascending chain condition in the definition of Dedekind domains is used in a similar way to the well-ordering principle in proving unique factorization of ideals in such domains.
answered 8 hours ago
MichaelGaudreauMichaelGaudreau
2842 silver badges6 bronze badges
answered 8 hours ago
MichaelGaudreauMichaelGaudreau
2842 silver badges6 bronze badges
answered 8 hours ago
MichaelGaudreauMichaelGaudreau
2842 silver badges6 bronze badges
answered 8 hours ago
MichaelGaudreauMichaelGaudreau
2842 silver badges6 bronze badges
2842 silver badges6 bronze badges
add a comment |
add a comment |
$begingroup$
In Pete Clark's notes on factorization, he calls this the "Lindemann–Zermelo proof" of the fundamental theorem of arithmetic, and he uses a similar idea to prove the well-known fact that if $R$ is a unique factorization domain, then so is $R[t]$. See Theorem 27.
answered 1 hour ago
Timothy ChowTimothy Chow
37k14 gold badges189 silver badges334 bronze badges
$endgroup$
add a comment |
$begingroup$
In Pete Clark's notes on factorization, he calls this the "Lindemann–Zermelo proof" of the fundamental theorem of arithmetic, and he uses a similar idea to prove the well-known fact that if $R$ is a unique factorization domain, then so is $R[t]$. See Theorem 27.
answered 1 hour ago
Timothy ChowTimothy Chow
37k14 gold badges189 silver badges334 bronze badges
$endgroup$
add a comment |
$begingroup$
In Pete Clark's notes on factorization, he calls this the "Lindemann–Zermelo proof" of the fundamental theorem of arithmetic, and he uses a similar idea to prove the well-known fact that if $R$ is a unique factorization domain, then so is $R[t]$. See Theorem 27.
answered 1 hour ago
Timothy ChowTimothy Chow
37k14 gold badges189 silver badges334 bronze badges
$endgroup$
In Pete Clark's notes on factorization, he calls this the "Lindemann–Zermelo proof" of the fundamental theorem of arithmetic, and he uses a similar idea to prove the well-known fact that if $R$ is a unique factorization domain, then so is $R[t]$. See Theorem 27.
answered 1 hour ago
Timothy ChowTimothy Chow
37k14 gold badges189 silver badges334 bronze badges
answered 1 hour ago
Timothy ChowTimothy Chow
37k14 gold badges189 silver badges334 bronze badges
answered 1 hour ago
Timothy ChowTimothy Chow
37k14 gold badges189 silver badges334 bronze badges
answered 1 hour ago
Timothy ChowTimothy Chow
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$begingroup$
Second-last line: "'the prime p is in the prime decomposition of (q-p)c and thus also at least of q-p or c. But we know that p∤c, whence p∣q-p." Are you using Euclid Lemma here, or am I missing something?
$endgroup$
– Francesco Polizzi
8 hours ago
$begingroup$
Since $n_0 < n$, by induction hypothesis $p$ must appear in the unique decomposition of $n_0 = (q-p)c$, which is the product of the unique decompositions of $q-p$ and $c$. It's true that the sentence is quite confusing.
$endgroup$
– Wille Liou
5 hours ago