# The Main Theorem

- Author:
- Ku, Yin Bon (Albert)

## Degree of a field extension

We already know that a quadratic extension has degree . Then how about iterated quadratic extension?
It turns out that the degree of any iterated quadratic extension must be a power of i.e. the number of parameters needed to describe all elements in an iterated quadratic extension is for some positive integer .
(

**Note**: Some linear algebra is needed to prove the above fact.)## Irreducible polynomials

Given a real number , if it is a root of a polynomial equation whose coefficients are in , we say that has the lowest degree. It is usually called the

**. Now we are going to simply the polynomial equation by factorising so as to lower its degree. Now, we may assume that** is algebraic over **irreducible polynomial of** over . The degree of such irreducible polynomial is called the**degree of** over .## The Main Theorem

Here is a very useful result about degrees:
If and be two field extensions over such that , then the degree of over is divisible by the degree of over .
Let be a real number that is algebraic over . If it is a constructible number, it must lie in an iterated quadratic extension over . Let be the field extension just large enough to contain and . It can be shown that the degree of over equals the degree of over . Therefore, and by above, the degree of over , which is a power of , is divisible by the degree of over . In other words, the degree of over is also a power of .
Now, we can rephrase the above important results as

**the main theorem**:**Given a real number** that is algebraic over . If the degree of over is not a power of , then is not a constructible number.
(**Note**: the detailed proof of the above theorem is beyond the scope of this course.)