How to describe the field extension?

Question

Consider the circle with radius $2\sqrt{2}$ centered at the origin and the line joining the points $\left(\frac{1}{2}, 0\right)$, $(4\sqrt{2}, \sqrt{2})$. The points of intersection of the line and circle have coordinates which lie in a field extension of $\mathbb{Q}$. Give a precise description of this field extension.