2022/060) If $x^2 - 3x +1 = 0$ and $x^{16} - k x^8 + 1 = 0$ then find k

Let $y^2- my + 1 = 0\cdots(1)$

so $y^2 = my -1$

or squaring both sides $y^4 = m^2y^2 - 2my + 1 = m^2y^2 - 2(y^2) + 1= (m^2-2)y^2$

or $y^4 = (m^2-2)y^2 + 1\cdots(2)$

putting y =x and m = 3 we get

$x^2 - 3x + 1$

and from (2) we have $x^4 - 7x^2 +1 = 0\cdots(3)$ 

putting $y = x^2$ and m = 7 in (1) we get (3) and from(2)  we get 

$x^8 - 47x^4 +1 = 9\cdots(4)$

  putting $y = x^4$ and m = 47 in (1) we get (4) and from(2)  we get