<s>[INST] Minimal polynomial of $\zeta+\zeta^{-1}$ 
Find the minimal polynomial of $\zeta+\zeta^{-1}\in \mathbb{Q}(\zeta)$ over $\mathbb{Q}$, where $\zeta$ is primitive $13^{th}$ root of unity.

All I know is that the minimal polynomial should be of degree 6.
My thougths
Usually, given an element, say $\sqrt{2}+\sqrt{3}$, to find the minimal polynomial, we take $\alpha=\sqrt{2}+\sqrt{3}$ and the square it and do further simplifications to get a linear combination of powers of $\alpha$ (polynomial in $\alpha$) equal to zero, If the resulting polynomial is irreducible, we say it is a minimal polynomial for the given element over the given field. However, for the element $\zeta+\zeta^{-1}$ this way is too complicated.
Could you suggest some other procedure (if any) or a hint to simplify the calculation?
 [/INST] We have the extensions
$$\mathbb{Q}\subset \mathbb{Q}( \zeta + \zeta^{-1}) \subset \mathbb{Q}(\zeta)$$
We have $[\mathbb{Q}(\zeta)\colon \mathbb{Q}( \zeta + \zeta^{-1})]\le 2$, and also $\ne 1$, since one is real, and the other is not. Therefore, the degree is $2$, and therefore
$$ [\mathbb{Q}( \zeta + \zeta^{-1})\colon \mathbb{Q}] = \frac{1}{2} [\mathbb{Q}(\zeta)\colon \mathbb{Q}]= \frac{\phi(n)}{2}$$
The minimal polynomial for $\zeta = \zeta_n$ is the cyclotomic polynomial of order $n$
$$\Phi_n(X)= \prod_{d \mid n} (X^{\frac{n}{d}}-1)^{\mu(d)} = \prod_{d \mid n} (1 + X + \cdots X^{\frac{n}{d} - 1})^{\mu(d)}$$
We see from the above that $\Phi_n(X)$ is palyndromic. We have therefore
$$\Phi_n(X) = X^{\frac{\phi(n)}{2}} Q( X+ \frac{1}{X})$$ where $\deg Q = \frac{\phi(n)}{2}$
In our case $n=13$,
$$\Phi_{13}(X) = \frac{X^{13}-1}{X-1}= (1+X + \cdots +X^{12}) = X^6( X^6 + \frac{1}{X^6} + X^{5}+ \frac{1}{X^5} + \cdots + X + \frac{1}{X} + 1)$$
We can use a CAS (WA) to express $\frac{x^{13}-1}{x^6(x-1)}$ as a polynomial in $x+ \frac{1}{x}$
$$\frac{x^{13}-1}{x^6(x-1)} =z^6 + z^5 - 5 z^4 - 4 z^3 + 6 z^2 + 3 z - 1$$
where $z= x+ \frac{1}{x}$. Therefore
$Q(z) = z^6 + z^5 - 5 z^4 - 4 z^3 + 6 z^2 + 3 z - 1$ is the minimal polynomial of $\zeta_{13} + \zeta_{13}^{-1}$, as one can check.
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