Prime norm ideals that are also principal
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Landau's prime number theorem gives us an asymptotic formula for counting the number of prime ideals of a number field $K$, with norm at most $X$.
I am interested in the the prime ideals with a prime norm. It is known (thanks to the comments) that prime ideals with prime norms are in fact the majority. But what about these ideals being principal? What faction of these do we expect to be principal?
number-theory prime-numbers
edited Nov 21 at 12:07
amWhy
191k27223439
asked May 12 '13 at 23:10
Siitd
413
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show 1 more comment
up vote
3
down vote
favorite
Landau's prime number theorem gives us an asymptotic formula for counting the number of prime ideals of a number field $K$, with norm at most $X$.
I am interested in the the prime ideals with a prime norm. It is known (thanks to the comments) that prime ideals with prime norms are in fact the majority. But what about these ideals being principal? What faction of these do we expect to be principal?
number-theory prime-numbers
edited Nov 21 at 12:07
amWhy
191k27223439
asked May 12 '13 at 23:10
Siitd
413
1
From en.wikipedia.org/wiki/Landau_prime_ideal_theorem: "... so that the prime ideal theorem is dominated by the ideals of norm a prime number." Therefore I guess the quantity you are looking for is still $X/log(X)$.
– Ralph
May 13 '13 at 0:08
1
The prime ideals whose norm is not a prime number have natural (or Dirichlet) density 0.
– KCd
May 13 '13 at 11:28
@KCd: And what about these prime ideals being principal? What fraction of these ideals do we expect to be principal?
– Siitd
May 13 '13 at 18:49
That would be $1/h$, where $h$ is the class number. This comes from class field theory.
– KCd
May 14 '13 at 3:17
1
@Siitd: Read about the Chebotarev density theorem and the Hilbert class field. It is deeper than a "lemma".
– KCd
May 14 '13 at 19:49
|
show 1 more comment
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Landau's prime number theorem gives us an asymptotic formula for counting the number of prime ideals of a number field $K$, with norm at most $X$.
I am interested in the the prime ideals with a prime norm. It is known (thanks to the comments) that prime ideals with prime norms are in fact the majority. But what about these ideals being principal? What faction of these do we expect to be principal?
number-theory prime-numbers
edited Nov 21 at 12:07
amWhy
191k27223439
asked May 12 '13 at 23:10
Siitd
413
Landau's prime number theorem gives us an asymptotic formula for counting the number of prime ideals of a number field $K$, with norm at most $X$.
I am interested in the the prime ideals with a prime norm. It is known (thanks to the comments) that prime ideals with prime norms are in fact the majority. But what about these ideals being principal? What faction of these do we expect to be principal?
number-theory prime-numbers
number-theory prime-numbers
edited Nov 21 at 12:07
amWhy
191k27223439
asked May 12 '13 at 23:10
Siitd
413
edited Nov 21 at 12:07
amWhy
191k27223439
asked May 12 '13 at 23:10
Siitd
413
edited Nov 21 at 12:07
amWhy
191k27223439
edited Nov 21 at 12:07
amWhy
191k27223439
edited Nov 21 at 12:07
amWhy
191k27223439
191k27223439
asked May 12 '13 at 23:10
Siitd
413
asked May 12 '13 at 23:10
Siitd
413
asked May 12 '13 at 23:10
Siitd
413
413
1
From en.wikipedia.org/wiki/Landau_prime_ideal_theorem: "... so that the prime ideal theorem is dominated by the ideals of norm a prime number." Therefore I guess the quantity you are looking for is still $X/log(X)$.
– Ralph
May 13 '13 at 0:08
1
The prime ideals whose norm is not a prime number have natural (or Dirichlet) density 0.
– KCd
May 13 '13 at 11:28
@KCd: And what about these prime ideals being principal? What fraction of these ideals do we expect to be principal?
– Siitd
May 13 '13 at 18:49
That would be $1/h$, where $h$ is the class number. This comes from class field theory.
– KCd
May 14 '13 at 3:17
1
@Siitd: Read about the Chebotarev density theorem and the Hilbert class field. It is deeper than a "lemma".
– KCd
May 14 '13 at 19:49
|
show 1 more comment
1
From en.wikipedia.org/wiki/Landau_prime_ideal_theorem: "... so that the prime ideal theorem is dominated by the ideals of norm a prime number." Therefore I guess the quantity you are looking for is still $X/log(X)$.
– Ralph
May 13 '13 at 0:08
1
The prime ideals whose norm is not a prime number have natural (or Dirichlet) density 0.
– KCd
May 13 '13 at 11:28
@KCd: And what about these prime ideals being principal? What fraction of these ideals do we expect to be principal?
– Siitd
May 13 '13 at 18:49
That would be $1/h$, where $h$ is the class number. This comes from class field theory.
– KCd
May 14 '13 at 3:17
1
@Siitd: Read about the Chebotarev density theorem and the Hilbert class field. It is deeper than a "lemma".
– KCd
May 14 '13 at 19:49
1
1
From en.wikipedia.org/wiki/Landau_prime_ideal_theorem: "... so that the prime ideal theorem is dominated by the ideals of norm a prime number." Therefore I guess the quantity you are looking for is still $X/log(X)$.
– Ralph
May 13 '13 at 0:08
From en.wikipedia.org/wiki/Landau_prime_ideal_theorem: "... so that the prime ideal theorem is dominated by the ideals of norm a prime number." Therefore I guess the quantity you are looking for is still $X/log(X)$.
– Ralph
May 13 '13 at 0:08
1
1
The prime ideals whose norm is not a prime number have natural (or Dirichlet) density 0.
– KCd
May 13 '13 at 11:28
The prime ideals whose norm is not a prime number have natural (or Dirichlet) density 0.
– KCd
May 13 '13 at 11:28
@KCd: And what about these prime ideals being principal? What fraction of these ideals do we expect to be principal?
– Siitd
May 13 '13 at 18:49
@KCd: And what about these prime ideals being principal? What fraction of these ideals do we expect to be principal?
– Siitd
May 13 '13 at 18:49
That would be $1/h$, where $h$ is the class number. This comes from class field theory.
– KCd
May 14 '13 at 3:17
That would be $1/h$, where $h$ is the class number. This comes from class field theory.
– KCd
May 14 '13 at 3:17
1
1
@Siitd: Read about the Chebotarev density theorem and the Hilbert class field. It is deeper than a "lemma".
– KCd
May 14 '13 at 19:49
@Siitd: Read about the Chebotarev density theorem and the Hilbert class field. It is deeper than a "lemma".
– KCd
May 14 '13 at 19:49
|
show 1 more comment
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1
From en.wikipedia.org/wiki/Landau_prime_ideal_theorem: "... so that the prime ideal theorem is dominated by the ideals of norm a prime number." Therefore I guess the quantity you are looking for is still $X/log(X)$.
– Ralph
May 13 '13 at 0:08
1
The prime ideals whose norm is not a prime number have natural (or Dirichlet) density 0.
– KCd
May 13 '13 at 11:28
@KCd: And what about these prime ideals being principal? What fraction of these ideals do we expect to be principal?
– Siitd
May 13 '13 at 18:49
That would be $1/h$, where $h$ is the class number. This comes from class field theory.
– KCd
May 14 '13 at 3:17
1
@Siitd: Read about the Chebotarev density theorem and the Hilbert class field. It is deeper than a "lemma".
– KCd
May 14 '13 at 19:49