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Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$.

Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1\quad\text{for some $w\in K$}.$$ Define $K$ to be green if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1\quad\text{for some $w\in \mathcal{O}_K$}.$$ (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 $$B_N:=|\{K:K\text{ is a blue (cyclic cubic) number field of conductor }<N\}|,$$ $$G_N:=|\{K:K\text{ is a green (cyclic cubic) number field of conductor }<N\}|,$$ what is $$ \lim_{N\to\infty} \frac{G_N}{B_N}? $$ (and does the limit exist?)

Question 3: Define $$\mathcal{G}:=\{f: K \text{ is green, where $K$ is a cyclic cubic number field of conductor $f$}\}.$$ What is $\mathcal{G}$ 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&degree=3&galois_group=C3&count=20). The code also explicitly gives the minimal polynomial of $w$. Here are the first few examples.

\begin{align*} 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 \end{align*}

The polynomials above prove that $\{7,9,13,19\}\subseteq\mathcal{G}$. Notice that for $f=31$, this polynomial implies $K$ of conductor $31$ is blue, but it may or may not be green.

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$.

Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1\quad\text{for some $w\in K$}.$$ Define $K$ to be green if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1\quad\text{for some $w\in \mathcal{O}_K$}.$$ (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 $$B_N:=\{K:K\text{ is a blue (cyclic cubic) number field of conductor }<N\},$$ $$G_N:=\{K:K\text{ is a green (cyclic cubic) number field of conductor }<N\},$$ what is $$ \lim_{N\to\infty} \frac{\#G_N}{\#B_N}? $$ (and does the limit exist?)

Question 3: Define $$\mathcal{G}:=\{f: K \text{ is green, where $K$ is a cyclic cubic number field of conductor $f$}\}.$$ What is $\mathcal{G}$ 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&degree=3&galois_group=C3&count=20). The code also explicitly gives the minimal polynomial of $w$. Here are the first few examples.

\begin{align*} 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 \end{align*}

The polynomials above prove that $\{7,9,13,19\}\subseteq\mathcal{G}$. Notice that for $f=31$, this polynomial implies $K$ of conductor $31$ is blue, but it may or may not be green.

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$.

Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1\quad\text{for some $w\in K$}.$$ Define $K$ to be green if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1\quad\text{for some $w\in \mathcal{O}_K$}.$$ (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 $$B_N:=\{K:K\text{ is a blue (cyclic cubic) number field of conductor }<N\},$$ $$G_N:=\{K:K\text{ is a green (cyclic cubic) number field of conductor }<N\},$$ what is $$ \lim_{N\to\infty} \frac{G_N}{B_N}? $$ (and does the limit exist?)

Question 3: Define $$\mathcal{G}:=\{f: K \text{ is green, where $K$ is a cyclic cubic number field of conductor $f$}\}.$$ What is $\mathcal{G}$ 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&degree=3&galois_group=C3&count=20). The code also explicitly gives the minimal polynomial of $w$. Here are the first few examples.

\begin{align*} 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 \end{align*}

The polynomials above prove that $\{7,9,13,19\}\subseteq\mathcal{G}$. Notice that for $f=31$, this polynomial implies $K$ of conductor $31$ is blue, but it may or may not be green.

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$.

Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1\quad\text{for some $w\in K$}.$$ Define $K$ to be green if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1\quad\text{for some $w\in \mathcal{O}_K$}.$$ (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 $$B_N:=|\{K:K\text{ is a blue (cyclic cubic) number field of conductor }<N\}|,$$ $$G_N:=|\{K:K\text{ is a green (cyclic cubic) number field of conductor }<N\}|,$$ what is $$ \lim_{N\to\infty} \frac{G_N}{B_N}? $$ (and does the limit exist?)

Question 3: Define $$\mathcal{G}:=\{f: K \text{ is green, where $K$ is a cyclic cubic number field of conductor $f$}\}.$$ What is $\mathcal{G}$ 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&degree=3&galois_group=C3&count=20). The code also explicitly gives the minimal polynomial of $w$. Here are the first few examples.

\begin{align*} 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 \end{align*}

The polynomials above prove that $\{7,9,13,19\}\subseteq\mathcal{G}$. Notice that for $f=31$, this polynomial implies $K$ of conductor $31$ is blue, but it may or may not be green.

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$.

Define $K$ to be "blue" if and only if there exists some element $w$ of $K$ such that $\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1$. Define $K$ to be "green" if and only if there exists some element $w$ of $\mathcal{O}_K$ such that $\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1$. (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 $B_N:=\{K:K\text{ is a blue (cyclic cubic) number field of conductor }<N\}$ and $G_N:=\{K:K\text{ is a green (cyclic cubic) number field of conductor }<N\}$, what is $$ \lim_{N\to\infty} \frac{G_N}{B_N}? $$ (and does the limit exist?)

Question 3: Define $\mathcal{G}:=\{f: K \text{ is green where $K:=$ a cyclic cubic number field of conductor $f$}\}$. What is $\mathcal{G}$ 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&degree=3&galois_group=C3&count=20). The code also explicitly gives the minimal polynomial of $w$. Here are the first few examples.

\begin{align*} 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 \end{align*}

The polynomials above prove that $\{7,9,13,19\}\subseteq\mathcal{G}$. Notice that for $f=31$, this polynomial implies $K$ of conductor $31$ is blue, but it may or may not be green.

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$.

Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1\quad\text{for some $w\in K$}.$$ Define $K$ to be green if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/\mathbb{Q}}(1-w) = -1\quad\text{for some $w\in \mathcal{O}_K$}.$$ (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 $$B_N:=\{K:K\text{ is a blue (cyclic cubic) number field of conductor }<N\},$$ $$G_N:=\{K:K\text{ is a green (cyclic cubic) number field of conductor }<N\},$$ what is $$ \lim_{N\to\infty} \frac{G_N}{B_N}? $$ (and does the limit exist?)

Question 3: Define $$\mathcal{G}:=\{f: K \text{ is green, where $K$ is a cyclic cubic number field of conductor $f$}\}.$$ What is $\mathcal{G}$ 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&degree=3&galois_group=C3&count=20). The code also explicitly gives the minimal polynomial of $w$. Here are the first few examples.

\begin{align*} 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 \end{align*}

The polynomials above prove that $\{7,9,13,19\}\subseteq\mathcal{G}$. Notice that for $f=31$, this polynomial implies $K$ of conductor $31$ is blue, but it may or may not be green.