Mixing
In which cyclic cubic number fields does there exist this type of unit?
Clash Royale CLAN TAG#URR8PPP
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Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $mathcalO_K$.
Define $K$ to be blue if and only if $$operatornameNorm_K/mathbbQ(w) = operatornameNorm_K/mathbbQ(1-w) = -1quadtextfor some $win K$.$$
Define $K$ to be green if and only if $$operatornameNorm_K/mathbbQ(w) = operatornameNorm_K/mathbbQ(1-w) = -1quadtextfor some $win mathcalO_K$.$$ (So green implies blue).
Question 1: Are all cyclic cubic number fields blue?
Question 2: What is the density of green number fields restricted to blue number fields? That is, defining $$B_N:=K:Ktext is a blue (cyclic cubic) number field of conductor <N,$$ $$G_N:=K:Ktext is a green (cyclic cubic) number field of conductor <N,$$ what is
$$
lim_Ntoinfty fracG_NB_N?
$$
(and does the limit exist?)
Question 3: Define $$mathcalG:=f: K text is green, where $K$ is a cyclic cubic number field of conductor $f$.$$ What is $mathcalG$ explicitly?
Remarks: I wrote some magma code that proved that $K$ is blue for all of the 1822 cubic cyclic number fields given from LMFDB (http://www.lmfdb.org/NumberField/start=0°ree=3&galois_group=C3&count=20). The code also explicitly gives the minimal polynomial of $w$. Here are the first few examples.
beginalign*
f=7, quad & t^3 - 2t^2 - t + 1
\
f=9, quad & t^3 - 3t + 1
\
f=13, quad & t^3 + t^2 - 4t + 1
\
f=19, quad & t^3 - 5t^2 + 2t + 1
\
f=31, quad & t^3 - (5/2)t^2 - (1/2)t + 1
endalign*
The polynomials above prove that $7,9,13,19subseteqmathcalG$. Notice that for $f=31$, this polynomial implies $K$ of conductor $31$ is blue, but it may or may not be green.
nt.number-theory algebraic-number-theory computational-number-theory number-fields
edited 27 mins ago
Joe Silverman
29.8k177155
asked 4 hours ago
Christine McMeekin
34519
add a comment |Â
up vote
7
down vote
favorite
Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $mathcalO_K$.
Define $K$ to be blue if and only if $$operatornameNorm_K/mathbbQ(w) = operatornameNorm_K/mathbbQ(1-w) = -1quadtextfor some $win K$.$$
Define $K$ to be green if and only if $$operatornameNorm_K/mathbbQ(w) = operatornameNorm_K/mathbbQ(1-w) = -1quadtextfor some $win mathcalO_K$.$$ (So green implies blue).
Question 1: Are all cyclic cubic number fields blue?
Question 2: What is the density of green number fields restricted to blue number fields? That is, defining $$B_N:=K:Ktext is a blue (cyclic cubic) number field of conductor <N,$$ $$G_N:=K:Ktext is a green (cyclic cubic) number field of conductor <N,$$ what is
$$
lim_Ntoinfty fracG_NB_N?
$$
(and does the limit exist?)
Question 3: Define $$mathcalG:=f: K text is green, where $K$ is a cyclic cubic number field of conductor $f$.$$ What is $mathcalG$ explicitly?
Remarks: I wrote some magma code that proved that $K$ is blue for all of the 1822 cubic cyclic number fields given from LMFDB (http://www.lmfdb.org/NumberField/start=0°ree=3&galois_group=C3&count=20). The code also explicitly gives the minimal polynomial of $w$. Here are the first few examples.
beginalign*
f=7, quad & t^3 - 2t^2 - t + 1
\
f=9, quad & t^3 - 3t + 1
\
f=13, quad & t^3 + t^2 - 4t + 1
\
f=19, quad & t^3 - 5t^2 + 2t + 1
\
f=31, quad & t^3 - (5/2)t^2 - (1/2)t + 1
endalign*
The polynomials above prove that $7,9,13,19subseteqmathcalG$. Notice that for $f=31$, this polynomial implies $K$ of conductor $31$ is blue, but it may or may not be green.
nt.number-theory algebraic-number-theory computational-number-theory number-fields
edited 27 mins ago
Joe Silverman
29.8k177155
asked 4 hours ago
Christine McMeekin
34519
2
+1 for the colourblind-friendly choice of colours.
– Chris Wuthrich
4 hours ago
After reading GNiklasch's comments, I think for the density question it may make more sense to restrict both $G_N$ and $B_N$ only to number fields in which 2 is inert/Q and then ask the limit of $G_N/B_N$ as $Ntoinfty$.
– Christine McMeekin
53 mins ago
Nice question. I'm going to edit a bit to improve readability, hope that's okay. Also, I'm sure you're already aware, but there's been lots of work on $winmathcal O_K$ such that $w$ and $1-w$ have norm 1, i.e., they're both units. Then $w$ is called an exceptional unit. So the $w$ in your blue case might be called very exceptional units!
– Joe Silverman
33 mins ago
It almost goes without saying, but $G_N$ and $B_N$ are the cardinalities of the two sets. (Too few characters for me to edit...)
– GNiklasch
4 mins ago
add a comment |Â
up vote
7
down vote
favorite
up vote
7
down vote
favorite
Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $mathcalO_K$.
Define $K$ to be blue if and only if $$operatornameNorm_K/mathbbQ(w) = operatornameNorm_K/mathbbQ(1-w) = -1quadtextfor some $win K$.$$
Define $K$ to be green if and only if $$operatornameNorm_K/mathbbQ(w) = operatornameNorm_K/mathbbQ(1-w) = -1quadtextfor some $win mathcalO_K$.$$ (So green implies blue).
Question 1: Are all cyclic cubic number fields blue?
Question 2: What is the density of green number fields restricted to blue number fields? That is, defining $$B_N:=K:Ktext is a blue (cyclic cubic) number field of conductor <N,$$ $$G_N:=K:Ktext is a green (cyclic cubic) number field of conductor <N,$$ what is
$$
lim_Ntoinfty fracG_NB_N?
$$
(and does the limit exist?)
Question 3: Define $$mathcalG:=f: K text is green, where $K$ is a cyclic cubic number field of conductor $f$.$$ What is $mathcalG$ explicitly?
Remarks: I wrote some magma code that proved that $K$ is blue for all of the 1822 cubic cyclic number fields given from LMFDB (http://www.lmfdb.org/NumberField/start=0°ree=3&galois_group=C3&count=20). The code also explicitly gives the minimal polynomial of $w$. Here are the first few examples.
beginalign*
f=7, quad & t^3 - 2t^2 - t + 1
\
f=9, quad & t^3 - 3t + 1
\
f=13, quad & t^3 + t^2 - 4t + 1
\
f=19, quad & t^3 - 5t^2 + 2t + 1
\
f=31, quad & t^3 - (5/2)t^2 - (1/2)t + 1
endalign*
The polynomials above prove that $7,9,13,19subseteqmathcalG$. Notice that for $f=31$, this polynomial implies $K$ of conductor $31$ is blue, but it may or may not be green.
nt.number-theory algebraic-number-theory computational-number-theory number-fields
edited 27 mins ago
Joe Silverman
29.8k177155
asked 4 hours ago
Christine McMeekin
34519
Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $mathcalO_K$.
Define $K$ to be blue if and only if $$operatornameNorm_K/mathbbQ(w) = operatornameNorm_K/mathbbQ(1-w) = -1quadtextfor some $win K$.$$
Define $K$ to be green if and only if $$operatornameNorm_K/mathbbQ(w) = operatornameNorm_K/mathbbQ(1-w) = -1quadtextfor some $win mathcalO_K$.$$ (So green implies blue).
Question 1: Are all cyclic cubic number fields blue?
Question 2: What is the density of green number fields restricted to blue number fields? That is, defining $$B_N:=K:Ktext is a blue (cyclic cubic) number field of conductor <N,$$ $$G_N:=K:Ktext is a green (cyclic cubic) number field of conductor <N,$$ what is
$$
lim_Ntoinfty fracG_NB_N?
$$
(and does the limit exist?)
Question 3: Define $$mathcalG:=f: K text is green, where $K$ is a cyclic cubic number field of conductor $f$.$$ What is $mathcalG$ explicitly?
Remarks: I wrote some magma code that proved that $K$ is blue for all of the 1822 cubic cyclic number fields given from LMFDB (http://www.lmfdb.org/NumberField/start=0°ree=3&galois_group=C3&count=20). The code also explicitly gives the minimal polynomial of $w$. Here are the first few examples.
beginalign*
f=7, quad & t^3 - 2t^2 - t + 1
\
f=9, quad & t^3 - 3t + 1
\
f=13, quad & t^3 + t^2 - 4t + 1
\
f=19, quad & t^3 - 5t^2 + 2t + 1
\
f=31, quad & t^3 - (5/2)t^2 - (1/2)t + 1
endalign*
The polynomials above prove that $7,9,13,19subseteqmathcalG$. Notice that for $f=31$, this polynomial implies $K$ of conductor $31$ is blue, but it may or may not be green.
nt.number-theory algebraic-number-theory computational-number-theory number-fields
nt.number-theory algebraic-number-theory computational-number-theory number-fields
edited 27 mins ago
Joe Silverman
29.8k177155
asked 4 hours ago
Christine McMeekin
34519
edited 27 mins ago
Joe Silverman
29.8k177155
asked 4 hours ago
Christine McMeekin
34519
edited 27 mins ago
Joe Silverman
29.8k177155
edited 27 mins ago
Joe Silverman
29.8k177155
edited 27 mins ago
Joe Silverman
29.8k177155
29.8k177155
asked 4 hours ago
Christine McMeekin
34519
asked 4 hours ago
Christine McMeekin
34519
asked 4 hours ago
Christine McMeekin
34519
34519
2
+1 for the colourblind-friendly choice of colours.
– Chris Wuthrich
4 hours ago
After reading GNiklasch's comments, I think for the density question it may make more sense to restrict both $G_N$ and $B_N$ only to number fields in which 2 is inert/Q and then ask the limit of $G_N/B_N$ as $Ntoinfty$.
– Christine McMeekin
53 mins ago
Nice question. I'm going to edit a bit to improve readability, hope that's okay. Also, I'm sure you're already aware, but there's been lots of work on $winmathcal O_K$ such that $w$ and $1-w$ have norm 1, i.e., they're both units. Then $w$ is called an exceptional unit. So the $w$ in your blue case might be called very exceptional units!
– Joe Silverman
33 mins ago
It almost goes without saying, but $G_N$ and $B_N$ are the cardinalities of the two sets. (Too few characters for me to edit...)
– GNiklasch
4 mins ago
add a comment |Â
2
+1 for the colourblind-friendly choice of colours.
– Chris Wuthrich
4 hours ago
After reading GNiklasch's comments, I think for the density question it may make more sense to restrict both $G_N$ and $B_N$ only to number fields in which 2 is inert/Q and then ask the limit of $G_N/B_N$ as $Ntoinfty$.
– Christine McMeekin
53 mins ago
Nice question. I'm going to edit a bit to improve readability, hope that's okay. Also, I'm sure you're already aware, but there's been lots of work on $winmathcal O_K$ such that $w$ and $1-w$ have norm 1, i.e., they're both units. Then $w$ is called an exceptional unit. So the $w$ in your blue case might be called very exceptional units!
– Joe Silverman
33 mins ago
It almost goes without saying, but $G_N$ and $B_N$ are the cardinalities of the two sets. (Too few characters for me to edit...)
– GNiklasch
4 mins ago
2
2
+1 for the colourblind-friendly choice of colours.
– Chris Wuthrich
4 hours ago
+1 for the colourblind-friendly choice of colours.
– Chris Wuthrich
4 hours ago
After reading GNiklasch's comments, I think for the density question it may make more sense to restrict both $G_N$ and $B_N$ only to number fields in which 2 is inert/Q and then ask the limit of $G_N/B_N$ as $Ntoinfty$.
– Christine McMeekin
53 mins ago
After reading GNiklasch's comments, I think for the density question it may make more sense to restrict both $G_N$ and $B_N$ only to number fields in which 2 is inert/Q and then ask the limit of $G_N/B_N$ as $Ntoinfty$.
– Christine McMeekin
53 mins ago
Nice question. I'm going to edit a bit to improve readability, hope that's okay. Also, I'm sure you're already aware, but there's been lots of work on $winmathcal O_K$ such that $w$ and $1-w$ have norm 1, i.e., they're both units. Then $w$ is called an exceptional unit. So the $w$ in your blue case might be called very exceptional units!
– Joe Silverman
33 mins ago
Nice question. I'm going to edit a bit to improve readability, hope that's okay. Also, I'm sure you're already aware, but there's been lots of work on $winmathcal O_K$ such that $w$ and $1-w$ have norm 1, i.e., they're both units. Then $w$ is called an exceptional unit. So the $w$ in your blue case might be called very exceptional units!
– Joe Silverman
33 mins ago
It almost goes without saying, but $G_N$ and $B_N$ are the cardinalities of the two sets. (Too few characters for me to edit...)
– GNiklasch
4 mins ago
It almost goes without saying, but $G_N$ and $B_N$ are the cardinalities of the two sets. (Too few characters for me to edit...)
– GNiklasch
4 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
5
down vote
The answer to question 1 is yes.
Pick any field element $x_1$ outside the rationals. Let $x_2$ and $x_3$ be its conjugates under the Galois group. Then the cross ratio $$w=frac(x_1-1)(x_2-x_3)(x_1-x_2)(1-x_3)$$ does the trick. (This is the same argument as in this answer.)
Note that these $w$ usually won't be algebraic integers, and they cannot be algebraic integers when the prime $2$ splits in $K$ (since $w$ and $1-w$ cannot both map to $1$ in the 2-element residue class field).
The simplest answer to question 3 I know of is that the set is what Daniel Shanks studied in The Simplest Cubic Fields. (This is a little white lie: Shanks had in fact started out by calling all cyclic cubic fields "simplest", and then proceeded to focus on a certain subset where the conductor is prime.) If $w$ is an algebraic unit in a cubic field satisfying the two norm conditions, then the minimal polynomial of $-w$ is necessarily of the Shanks form $t^3-at^2 -(a+3)t-1$ for some rational integer $a$, and passing to $1/w$ if necessary one can assume $age -1$. So you can enumerate all "green" fields by letting $a$ vary (with only a few repetitions, but that's a deep result) and computing conductors (since $w$ won't necessarily generate the full ring of integers: $a=1259$ being the most spectacular case).
I can't answer question 2 offhand.
answered 2 hours ago
GNiklasch
1,735717
Thanks! Perhaps then in the density question, I should consider only those cyclic cubic number fields in which 2 is inert/Q. From the LMFDB, I computed that f is green for at least 89 of the 810 conductors st. 2 is inert in K, cyclic cubic of conductor f.
– Christine McMeekin
1 hour ago
Regarding your response to question 3, I think you mean $t^3+at^2−(a+3)t+1$. I've basically already done this except backwards; the code I wrote starts with $f$ and then computes $a$ instead of the other way around. If $a$ turns out to be an integer then I know $K$ is green, but if not then I'm not sure I can say whether $K$ is green or not. My code uses ideas from ``On Cyclic Cubic Fields" by Ennola and Turunen. I will continue to think about this.
– Christine McMeekin
59 mins ago
Fixed the wrong sign (thanks for pointing it out!).- Given a number field by a defining polynomial, it is always possible (at least in principle) to compute the set of all solutions to the unit equation $x+y=1$ in $mathcalO_K^times$, also known (following Nagell) as the set of exceptional units of $K$; in the cubic case it can be done "the wrong way round" by reducing to a Thue equation (even though computer algebra systems might internally reduce the Thue equation to an ($S$-)unit equation).
– GNiklasch
18 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
The answer to question 1 is yes.
Pick any field element $x_1$ outside the rationals. Let $x_2$ and $x_3$ be its conjugates under the Galois group. Then the cross ratio $$w=frac(x_1-1)(x_2-x_3)(x_1-x_2)(1-x_3)$$ does the trick. (This is the same argument as in this answer.)
Note that these $w$ usually won't be algebraic integers, and they cannot be algebraic integers when the prime $2$ splits in $K$ (since $w$ and $1-w$ cannot both map to $1$ in the 2-element residue class field).
The simplest answer to question 3 I know of is that the set is what Daniel Shanks studied in The Simplest Cubic Fields. (This is a little white lie: Shanks had in fact started out by calling all cyclic cubic fields "simplest", and then proceeded to focus on a certain subset where the conductor is prime.) If $w$ is an algebraic unit in a cubic field satisfying the two norm conditions, then the minimal polynomial of $-w$ is necessarily of the Shanks form $t^3-at^2 -(a+3)t-1$ for some rational integer $a$, and passing to $1/w$ if necessary one can assume $age -1$. So you can enumerate all "green" fields by letting $a$ vary (with only a few repetitions, but that's a deep result) and computing conductors (since $w$ won't necessarily generate the full ring of integers: $a=1259$ being the most spectacular case).
I can't answer question 2 offhand.
answered 2 hours ago
GNiklasch
1,735717
Thanks! Perhaps then in the density question, I should consider only those cyclic cubic number fields in which 2 is inert/Q. From the LMFDB, I computed that f is green for at least 89 of the 810 conductors st. 2 is inert in K, cyclic cubic of conductor f.
– Christine McMeekin
1 hour ago
Regarding your response to question 3, I think you mean $t^3+at^2−(a+3)t+1$. I've basically already done this except backwards; the code I wrote starts with $f$ and then computes $a$ instead of the other way around. If $a$ turns out to be an integer then I know $K$ is green, but if not then I'm not sure I can say whether $K$ is green or not. My code uses ideas from ``On Cyclic Cubic Fields" by Ennola and Turunen. I will continue to think about this.
– Christine McMeekin
59 mins ago
Fixed the wrong sign (thanks for pointing it out!).- Given a number field by a defining polynomial, it is always possible (at least in principle) to compute the set of all solutions to the unit equation $x+y=1$ in $mathcalO_K^times$, also known (following Nagell) as the set of exceptional units of $K$; in the cubic case it can be done "the wrong way round" by reducing to a Thue equation (even though computer algebra systems might internally reduce the Thue equation to an ($S$-)unit equation).
– GNiklasch
18 mins ago
add a comment |Â
up vote
5
down vote
The answer to question 1 is yes.
Pick any field element $x_1$ outside the rationals. Let $x_2$ and $x_3$ be its conjugates under the Galois group. Then the cross ratio $$w=frac(x_1-1)(x_2-x_3)(x_1-x_2)(1-x_3)$$ does the trick. (This is the same argument as in this answer.)
Note that these $w$ usually won't be algebraic integers, and they cannot be algebraic integers when the prime $2$ splits in $K$ (since $w$ and $1-w$ cannot both map to $1$ in the 2-element residue class field).
The simplest answer to question 3 I know of is that the set is what Daniel Shanks studied in The Simplest Cubic Fields. (This is a little white lie: Shanks had in fact started out by calling all cyclic cubic fields "simplest", and then proceeded to focus on a certain subset where the conductor is prime.) If $w$ is an algebraic unit in a cubic field satisfying the two norm conditions, then the minimal polynomial of $-w$ is necessarily of the Shanks form $t^3-at^2 -(a+3)t-1$ for some rational integer $a$, and passing to $1/w$ if necessary one can assume $age -1$. So you can enumerate all "green" fields by letting $a$ vary (with only a few repetitions, but that's a deep result) and computing conductors (since $w$ won't necessarily generate the full ring of integers: $a=1259$ being the most spectacular case).
I can't answer question 2 offhand.
answered 2 hours ago
GNiklasch
1,735717
Thanks! Perhaps then in the density question, I should consider only those cyclic cubic number fields in which 2 is inert/Q. From the LMFDB, I computed that f is green for at least 89 of the 810 conductors st. 2 is inert in K, cyclic cubic of conductor f.
– Christine McMeekin
1 hour ago
Regarding your response to question 3, I think you mean $t^3+at^2−(a+3)t+1$. I've basically already done this except backwards; the code I wrote starts with $f$ and then computes $a$ instead of the other way around. If $a$ turns out to be an integer then I know $K$ is green, but if not then I'm not sure I can say whether $K$ is green or not. My code uses ideas from ``On Cyclic Cubic Fields" by Ennola and Turunen. I will continue to think about this.
– Christine McMeekin
59 mins ago
Fixed the wrong sign (thanks for pointing it out!).- Given a number field by a defining polynomial, it is always possible (at least in principle) to compute the set of all solutions to the unit equation $x+y=1$ in $mathcalO_K^times$, also known (following Nagell) as the set of exceptional units of $K$; in the cubic case it can be done "the wrong way round" by reducing to a Thue equation (even though computer algebra systems might internally reduce the Thue equation to an ($S$-)unit equation).
– GNiklasch
18 mins ago
add a comment |Â
up vote
5
down vote
up vote
5
down vote
The answer to question 1 is yes.
Pick any field element $x_1$ outside the rationals. Let $x_2$ and $x_3$ be its conjugates under the Galois group. Then the cross ratio $$w=frac(x_1-1)(x_2-x_3)(x_1-x_2)(1-x_3)$$ does the trick. (This is the same argument as in this answer.)
Note that these $w$ usually won't be algebraic integers, and they cannot be algebraic integers when the prime $2$ splits in $K$ (since $w$ and $1-w$ cannot both map to $1$ in the 2-element residue class field).
The simplest answer to question 3 I know of is that the set is what Daniel Shanks studied in The Simplest Cubic Fields. (This is a little white lie: Shanks had in fact started out by calling all cyclic cubic fields "simplest", and then proceeded to focus on a certain subset where the conductor is prime.) If $w$ is an algebraic unit in a cubic field satisfying the two norm conditions, then the minimal polynomial of $-w$ is necessarily of the Shanks form $t^3-at^2 -(a+3)t-1$ for some rational integer $a$, and passing to $1/w$ if necessary one can assume $age -1$. So you can enumerate all "green" fields by letting $a$ vary (with only a few repetitions, but that's a deep result) and computing conductors (since $w$ won't necessarily generate the full ring of integers: $a=1259$ being the most spectacular case).
I can't answer question 2 offhand.
answered 2 hours ago
GNiklasch
1,735717
The answer to question 1 is yes.
Pick any field element $x_1$ outside the rationals. Let $x_2$ and $x_3$ be its conjugates under the Galois group. Then the cross ratio $$w=frac(x_1-1)(x_2-x_3)(x_1-x_2)(1-x_3)$$ does the trick. (This is the same argument as in this answer.)
Note that these $w$ usually won't be algebraic integers, and they cannot be algebraic integers when the prime $2$ splits in $K$ (since $w$ and $1-w$ cannot both map to $1$ in the 2-element residue class field).
The simplest answer to question 3 I know of is that the set is what Daniel Shanks studied in The Simplest Cubic Fields. (This is a little white lie: Shanks had in fact started out by calling all cyclic cubic fields "simplest", and then proceeded to focus on a certain subset where the conductor is prime.) If $w$ is an algebraic unit in a cubic field satisfying the two norm conditions, then the minimal polynomial of $-w$ is necessarily of the Shanks form $t^3-at^2 -(a+3)t-1$ for some rational integer $a$, and passing to $1/w$ if necessary one can assume $age -1$. So you can enumerate all "green" fields by letting $a$ vary (with only a few repetitions, but that's a deep result) and computing conductors (since $w$ won't necessarily generate the full ring of integers: $a=1259$ being the most spectacular case).
I can't answer question 2 offhand.
answered 2 hours ago
GNiklasch
1,735717
edited 29 mins ago
answered 2 hours ago
GNiklasch
1,735717
answered 2 hours ago
GNiklasch
1,735717
answered 2 hours ago
GNiklasch
1,735717
1,735717
Thanks! Perhaps then in the density question, I should consider only those cyclic cubic number fields in which 2 is inert/Q. From the LMFDB, I computed that f is green for at least 89 of the 810 conductors st. 2 is inert in K, cyclic cubic of conductor f.
– Christine McMeekin
1 hour ago
Regarding your response to question 3, I think you mean $t^3+at^2−(a+3)t+1$. I've basically already done this except backwards; the code I wrote starts with $f$ and then computes $a$ instead of the other way around. If $a$ turns out to be an integer then I know $K$ is green, but if not then I'm not sure I can say whether $K$ is green or not. My code uses ideas from ``On Cyclic Cubic Fields" by Ennola and Turunen. I will continue to think about this.
– Christine McMeekin
59 mins ago
Fixed the wrong sign (thanks for pointing it out!).- Given a number field by a defining polynomial, it is always possible (at least in principle) to compute the set of all solutions to the unit equation $x+y=1$ in $mathcalO_K^times$, also known (following Nagell) as the set of exceptional units of $K$; in the cubic case it can be done "the wrong way round" by reducing to a Thue equation (even though computer algebra systems might internally reduce the Thue equation to an ($S$-)unit equation).
– GNiklasch
18 mins ago
add a comment |Â
Thanks! Perhaps then in the density question, I should consider only those cyclic cubic number fields in which 2 is inert/Q. From the LMFDB, I computed that f is green for at least 89 of the 810 conductors st. 2 is inert in K, cyclic cubic of conductor f.
– Christine McMeekin
1 hour ago
Regarding your response to question 3, I think you mean $t^3+at^2−(a+3)t+1$. I've basically already done this except backwards; the code I wrote starts with $f$ and then computes $a$ instead of the other way around. If $a$ turns out to be an integer then I know $K$ is green, but if not then I'm not sure I can say whether $K$ is green or not. My code uses ideas from ``On Cyclic Cubic Fields" by Ennola and Turunen. I will continue to think about this.
– Christine McMeekin
59 mins ago
Fixed the wrong sign (thanks for pointing it out!).- Given a number field by a defining polynomial, it is always possible (at least in principle) to compute the set of all solutions to the unit equation $x+y=1$ in $mathcalO_K^times$, also known (following Nagell) as the set of exceptional units of $K$; in the cubic case it can be done "the wrong way round" by reducing to a Thue equation (even though computer algebra systems might internally reduce the Thue equation to an ($S$-)unit equation).
– GNiklasch
18 mins ago
Thanks! Perhaps then in the density question, I should consider only those cyclic cubic number fields in which 2 is inert/Q. From the LMFDB, I computed that f is green for at least 89 of the 810 conductors st. 2 is inert in K, cyclic cubic of conductor f.
– Christine McMeekin
1 hour ago
Thanks! Perhaps then in the density question, I should consider only those cyclic cubic number fields in which 2 is inert/Q. From the LMFDB, I computed that f is green for at least 89 of the 810 conductors st. 2 is inert in K, cyclic cubic of conductor f.
– Christine McMeekin
1 hour ago
Regarding your response to question 3, I think you mean $t^3+at^2−(a+3)t+1$. I've basically already done this except backwards; the code I wrote starts with $f$ and then computes $a$ instead of the other way around. If $a$ turns out to be an integer then I know $K$ is green, but if not then I'm not sure I can say whether $K$ is green or not. My code uses ideas from ``On Cyclic Cubic Fields" by Ennola and Turunen. I will continue to think about this.
– Christine McMeekin
59 mins ago
Regarding your response to question 3, I think you mean $t^3+at^2−(a+3)t+1$. I've basically already done this except backwards; the code I wrote starts with $f$ and then computes $a$ instead of the other way around. If $a$ turns out to be an integer then I know $K$ is green, but if not then I'm not sure I can say whether $K$ is green or not. My code uses ideas from ``On Cyclic Cubic Fields" by Ennola and Turunen. I will continue to think about this.
– Christine McMeekin
59 mins ago
Fixed the wrong sign (thanks for pointing it out!).- Given a number field by a defining polynomial, it is always possible (at least in principle) to compute the set of all solutions to the unit equation $x+y=1$ in $mathcalO_K^times$, also known (following Nagell) as the set of exceptional units of $K$; in the cubic case it can be done "the wrong way round" by reducing to a Thue equation (even though computer algebra systems might internally reduce the Thue equation to an ($S$-)unit equation).
– GNiklasch
18 mins ago
Fixed the wrong sign (thanks for pointing it out!).- Given a number field by a defining polynomial, it is always possible (at least in principle) to compute the set of all solutions to the unit equation $x+y=1$ in $mathcalO_K^times$, also known (following Nagell) as the set of exceptional units of $K$; in the cubic case it can be done "the wrong way round" by reducing to a Thue equation (even though computer algebra systems might internally reduce the Thue equation to an ($S$-)unit equation).
– GNiklasch
18 mins ago
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2
+1 for the colourblind-friendly choice of colours.
– Chris Wuthrich
4 hours ago
After reading GNiklasch's comments, I think for the density question it may make more sense to restrict both $G_N$ and $B_N$ only to number fields in which 2 is inert/Q and then ask the limit of $G_N/B_N$ as $Ntoinfty$.
– Christine McMeekin
53 mins ago
Nice question. I'm going to edit a bit to improve readability, hope that's okay. Also, I'm sure you're already aware, but there's been lots of work on $winmathcal O_K$ such that $w$ and $1-w$ have norm 1, i.e., they're both units. Then $w$ is called an exceptional unit. So the $w$ in your blue case might be called very exceptional units!
– Joe Silverman
33 mins ago
It almost goes without saying, but $G_N$ and $B_N$ are the cardinalities of the two sets. (Too few characters for me to edit...)
– GNiklasch
4 mins ago