This is a follow up to a question that I posed a few days ago here: The product of n radii in an ellipse
Suppose we have an ellipse $x^2/a^2 + y^2/b^2 = 1$ (centered at the origin). Let $n>4$. There are $n$ rays going out of the origin, at angles $0, 2\pi/n, 4\pi/n, 6\pi/n,...,2\pi(n-1)/n$. Let $(x_0,y_0),...,(x_{n-1},y_{n-1})$ be the points of intersection of the rays and the ellipse. The product from $k=0$ to $n-1$ of ${x_k}^2 + {y_k}^2$ is equal to one. Can $a$ and $b$ be rational if $a \neq b$?
It was discovered here that the answer is no, because Fermat's Last Theorem is true. Notice that the points of intersection of the rays and the ellipse, $(x_k,y_k)=x_k+{y_k}i$, generate a subfield of the complex numbers, since the ellipse is symmetric about the real line and the product of the points is one and the sum of the points is zero. (If one takes the set of all elements that can be expressed as a combination of additions and multiplications of the points of intersection of the rays and the ellipse, this set will be a subfield of the complex numbers, since zero and one are in this set.)
Conjecture: If $a,b$ are rational and $a \neq b$, the set of all complex numbers that can be expressed as a combination of additions and multiplications of the points of intersection of the rays and the ellipse, $(x_k,y_k)=x_k+{y_k}i$, cannot be a subfield of the complex numbers.
Prove this conjecture.
Craig
