Finding minimal polynomial of generator of quadratic subfield of cyclotomic extension $\mathbb{Q}(\xi_{17})$

Question

I will prove that $K_2=\mathbb{Q}(\sqrt{17})$ skipping a few 'trivial' steps to keep the post short. There is a theorem that says that Gal$(\mathbb{Q}(\xi_{17}))\cong\left(\mathbb{Z}_{17}\right)^{\times}\cong\mathbb{Z}_{16}$. We claim that $3$ is a generator of the multiplicative group $\left(\mathbb{Z}_{17}\right)^{\times}$ of order $16$; this I have shown.

Thus $\langle3\rangle=\left(\mathbb{Z}_{17}\right)^{\times}$. We define the following maps: \begin{align*} \text{Gal}(\mathbb{Q}(\xi_{17}))&\to\left(\mathbb{Z}_{17}\right)^{\times},\\ \left(\sigma_k:\xi_{17}\mapsto\xi_{17}^k\right)&\mapsto k\text{ mod }17\\ \varphi(n):3&\mapsto3^n\text{ mod }17. \end{align*} We also know that there are subfields $K_d$ of $\mathbb{Q}(\xi_{17})$ such that $[K_d:\mathbb{Q}]=d$ with $d\in\{1,2,4,8\}$. We now list subgroups $A,B,C$ of order $2,4,8$ respectively. \begin{align*} A&=\{0,8\}\\ B&=\{0,4,8,12\}\\ C&=\{0,2,4,6,8,10,12,14\}\\ \end{align*} We now apply $\varphi$ to the three groups $A,B,C$: \begin{align*} A'&=\varphi({A})=\{1,-1\}\\ B'&=\varphi({B})=\{1,-1,4,-4\}\\ C'&=\varphi({C})=\{1,-1,2,-2,4,-4,8,-8\}\\ \end{align*} We define \begin{align*} \tau_{A}&=\sum_{k\in A'}\xi_{17}^k\\ \tau_{B}&=\sum_{k\in B'}\xi_{17}^k\\ \tau_{C}&=\sum_{k\in C'}\xi_{17}^k. \end{align*} The subfields of $\mathbb{Q}(\xi_{17})$ are $\mathbb{Q}(\tau_A),\mathbb{Q}(\tau_B),\mathbb{Q}(\tau_C)$. The minimal polynomials are \begin{align*} m_{\tau_Z}=\prod_{k\in G}(x-\sigma_k(\tau_{Z})) \end{align*} where $Z\in\{A,B,C\}$ and $G$ is a set of length $\#Z$ which does contain $\#Z$ elements from $\mathbb{Z}_{16}-Z$. This last sentence above is where I can not verify what I'm saying and I don't what should be correct. We compute: \begin{align*} m_{\tau_C}=\prod_{k\in \{3,-3\}}(x-\sigma_k(\tau_{C}))=x^2+x-4. \end{align*} Its roots are $\frac{-1\pm\sqrt{17}}{2}$ and $m_{\tau_C}$ is irreducible since it has no rational roots (only linear factors would have been possible), so $K_2=\mathbb{Q}(\tau_C)=\mathbb{Q}(\sqrt{17})$.

So, partly I know what I'm doing but at the point where I try to compute the minimal polynomials, I have no idea which indices to take in the product... The minimal polynomial I just bluffed by letting Mathematica compute it via the minimal polynomial finder on Wolfram online.

Extra: My Mathematica attempt reads the following

xi = Exp[2*Pi*I/17];
sigma[t_, n_] :=t^n;
tauC = xi^1 + xi^(-1) + xi^2 + xi^(-2) + xi^4 + xi^(-4) + xi^8 + xi^9;
Product[(x - sigma[xi, i]), {i, 2}] // Expand // N // Rationalize

(0.445738 + 0.895163 I) - (1.67148 + 1.03494 I) x + x^2 

Btw, the minimal polynomimal of degree $6$ of the generator of the subfield $K_6$ of the cyclotomic field $\mathbb{Q}(\xi_{13})$ I did find this way taking the product from $1$ to $6$. Replacing $13$ by $17$ gets me trouble. How is that possible?

Any help or suggestion is appreciated!

Answer 1

What you seem to be having trouble with is the following.

You have correctly verified that the big Galois group $G=Gal(K_{16}/K_1)$ is of order $16$, and generated by $\sigma_3$. It has subgroups $$ \begin{aligned} Gal(K_{16}/K_2)&=\langle\sigma_3^2\rangle=\langle \sigma_9\rangle&=C',\\ Gal(K_{16}/K_4)&=\langle\sigma_3^4\rangle=\langle\sigma_{-4}\rangle&=B',\\ Gal(K_{16}/K_8)&=\langle\sigma_3^8\rangle=\langle\sigma_{-1}\rangle&=A'. \end{aligned} $$ As $G$ is abelian, these are all normal subgroups of $G$. You get the Galois groups for the extensions $K_d/K_1,d\in\{2,4,8\}$, as quotient groups of $G$. So $$ \begin{aligned} Gal(K_2/K_1)&=G/C',\\ Gal(K_4/K_1)&=G/B',\\ Gal(K_8/K_1)&=G/A'. \end{aligned} $$ Because $G$ is generated by $\sigma_3$, these are generated by the cosets $\sigma_3C'$, $\sigma_3B'$ and $\sigma_3A'$ respectively.

Therefore the sought after minimal polynomials are $$ \begin{aligned} m_{\tau_A}(x)=\prod_{j=0}^{8-1}\left(x-\sigma_3^j(\tau_A)\right),\\ m_{\tau_B}(x)=\prod_{j=0}^{4-1}\left(x-\sigma_3^j(\tau_B)\right),&\ \text{and}\\ m_{\tau_C}(x)=\prod_{j=0}^{2-1}\left(x-\sigma_3^j(\tau_C)\right).\\ \end{aligned} $$ You stop the products just before you reach the lowest positive power of $\sigma_3$ that belongs to the relevant subgroup. After all, the zeros of the minimal polynomials are simple, $\tau_A$ is invariant under the powers of $\sigma_3^8$, $\tau_B$ under the powers of $\sigma_3^4$, and $\tau_C$ already under $\sigma_3^2$. In yet other words, the powers of $\sigma_3$ in these product formulas range over the relevant quotient group.

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The following Mathematica snippet does it

xi = Exp[2 Pi I/17];
tau[d_, j_] := Sum[xi^(3^(d k + j)), {k, 1, 16/d}];
minpol[d_] := N[Expand[Product[x - tau[d, j], {j, 0, d - 1}]]];

Here $d$ must be a factor of $16$ for the formulas to make sense. My definition of tau[d_,j_] calculates $\sigma_3^j(\tau_X)$. Here $d=2$ gives $X=C$, $d=4$ gives $X=B$ and $d=8$ gives $X=A$. The point is that $Gal(K_d/\Bbb{Q})$ consists of the restrictions of powers of $\sigma_3^j=\sigma_{3^j}$ with $j$ ranging from $0$ to $d-1$.

The use of $3^j$ comes from the isomorphism $\phi:\Bbb{Z}_{16}\to Gal(K_{16}/\Bbb{Q})$ that maps $j$ to $\phi(j)=\sigma_3^j=\sigma_{3^j}$. My definition for tau[d_,j_] applies this to the (additive) subgroup $\langle d\rangle\le\Bbb{Z}_{16}, d=2,4,8$. We get $3^j$ instead of $3j$ as the index of $\sigma$ because $\phi$ maps an additive structure to a multiplicative one. If you find this confusing, have Mathematica spell out tau[d,j] for various choices of $d\in\{2,4,8\}, j=0,1,\ldots,d-1$, and see which seventeenth root of unity appear in which conjugate. I'm sure you'll get the hang of it!

Anyway, this gives the predicted (-4. + 0. I) + (1. + 0. I) x + x^2 as minpol[2]. Leaving you the fun to check out minpol[4] and minpol[8].