Explaining the impossibility of squaring the circle mathematically can someone please explain why squaring the circle was proved impossble by the way that they were attempting to solve it?
 A: The result relies fundamentally on the theory of field extensions of $\mathbb{Q}$.  Basically, what you need to observe is that a point in the plane is constructible in one step if and only if it lies in $K_1^2\subset\mathbb{R}^2$ where $K$ is a field extension of $K_0=\mathbb{Q}$ obtained by adjoining roots of a quadratic polynomial with coefficients in $K_0=\mathbb{Q}$.  Then you show that a point is constructible in $n$-steps if and only if it lies in $K_n^2\subset\mathbb{R}^2$ where $K_n$ is a field obtained by adjoining a root of a quadratic polynomial with coefficients in some $K_{n-1}$.  That is, we just iterate the process of adjoining roots of a quadratic polynomial to our current field $n$ times.
The important point is that whenever you adjoin a root of a degree two polynomial in this way, you create a field extension whose degree is at most $2$ over the base field.  So any field obtained via this procedure is a finite field extension of $\mathbb{Q}$.
Then once you know that $\pi$ is trancendental over $\mathbb{Q}$, it follows that $\pi$ doesn't lie in any field extension of $\mathbb{Q}$ with finite degree.  Therefore, the point $(0,\pi)$ cannot be constructed in finitely many steps with a ruler and compass.
