# On using field extensions to prove the impossiblity of a straightedge and compass construction

Let $z \in \mathbb{C}$. Consider the following statements:

1. The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$.
2. There is a field extension $K / \mathbb{Q}$ which has a tower of subextensions, each one of degree 2 over the next, and such that $z \in K$
3. The field extension $\mathbb{Q}(z) / \mathbb{Q}$ has a tower of subextensions, each one of degree 2 over the next.

The usual way to prove that a geometric construction is impossible is to use that 1 and 2 are equivalent. My question is: are 2 and 3 equivalent? At first sight this looked like it was going to be true and elementary, but I could not prove it or find a counterexample.

• Michael Artin covers this in depth in chapter 13 of his Algebra book. – Harry Gindi Feb 10 '10 at 22:31

• The lemma works for $p$-groups, where $p$ is any prime. – Chandan Singh Dalawat Feb 11 '10 at 4:10