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][1]). 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.


  [1]: http://www.lmfdb.org/NumberField/?start=0&degree=3&galois_group=C3&count=20