The Main Theorem

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 is algebraic over . Now we are going to simply the polynomial equation by factorising so as to lower its degree. Now, we may assume that  has the lowest degree. It is usually called the 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.)