<s>[INST] Find the minimum $n$ such that $x^3-\frac{2\sqrt2+1}{2}x^2+\frac{2\sqrt2+1}{2}x-1\mid x^n-1$ Set $f(x)= x^3-\frac{2\sqrt2+1}{2}x^2+\frac{2\sqrt2+1}{2}x-1$. Next I found the roots of this polynomial that $x=1$ and $\frac{1}{2}(\frac{2\sqrt2 -1}{2} \pm i \frac{\sqrt7+1}{2})$. Here the $n$ is a natural number.
But I can't proceed the next step. What should I do to find the minimal value of the $n$ without any just calculation (long division)? The method not finding roots also welcomed.
 [/INST] The question is wrong, in that the $f$ given in the question does not divide $x^n -1$ for any positive $n$. Let $f$ be a monic cubic of the form
$$f(x) = (x-1)(x^2-2cx+1)=x^3 - (1+2c)x^2 + (1+2c)x - 1$$ such that $c$ is quadratic over $\mathbb{Q}$ and such that $f\mid(x^n-1)$.
Since $c$ is quadratic over $\mathbb{Q}$, the corresponding roots of unity are algebraic of degree $4$ over $\mathbb{Q}$. But by the totient function formula for the degree of cyclotomic polynomials, such is the case only if the roots are $n$th roots of unity for $n=5,8,10,12$, so that the only polynomials of the prescribed form that divide some $x^n-1$ are
$$\begin{align}
f_{5\pm}(x)&=(x-1)(x^2\mp(\tfrac{\sqrt{5}-1}{2})x+1) \\
f_{8\pm}(x) &= (x-1)(x^2\mp\sqrt{2}x+1) \\
f_{10\pm}(x) &= (x-1)(x^2\mp(\tfrac{\sqrt{5} + 1}{2})x+1) \\
f_{12\pm}(x) &= (x-1)(x^2 \mp\sqrt{3}x+1)\text{.}
\end{align}$$
But the $f$ of the problem statement
$$f(x)=(x-1)(x^2-(\sqrt{2}-\tfrac{1}{2})x + 1)$$
is not on this list.
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