Field Extensions

About field extensions

As we know, every Euclidean construction starts with  and . Therefore, by definition, and are constructible numbers i.e. they are in . Since is a field, it must contain all rational numbers i.e. is a subset of . Now, we obtain  by constructing the diagonal of a unit square. Therefore, is constructible i.e. it is in . Obviously, is not a rational number. We can "mix" it with rational numbers using rational operations to generate a field just large enough to contain and . It is called the field extension of by adjoining , denoted by .

Quadratic extensions

It turns out that it is very easy to describe the real numbers in : It must be of the form , where and are in . For example, we consider in . We can easily bring it to the form by multiplying both the numerator and denominator by . Note that there are two free parameters and that describe that whole field extension . We say that the degree of the field extension is 2.  Sometimes it is called a quadratic extension over . More generally, given any field , if is in such that is not in , then is a quadratic extension over and all numbers in such quadratic extension are in the form , where and are in . (Note: If you know a bit linear algebra, it is not hard to see that a quadratic extension can be regarded as a two-dimensional vector space over .)

About constructible numbers

We already know that we can do rational operations and taking square root by Euclidean constructions. A natural question to answer is: Can Euclidean constructions do more? It turns out that with the help of coordinate geometry, it can be shown that these are pretty much all Euclidean constructions can do and nothing more... Take any constructible number , the point can be produced by a Euclidean construction. Suppose such construction has a number of steps. We can associate each step with a field that contains the coordinates of points constructed in such step as follows: Initially, we are given the points and . So the field that just contains and is . Suppose an intersection point obtained in a certain construction step that has coordinate where in and it is not in (for example, ). Then we consider the quadratic field extension that can contain the coordinates of newly-constructed point. Subsequently, if another intersection point obtained that has coordinate where in and it is not in , we consider the quadratic field extension that can contain the coordinates of newly-constructed point. This procedure continues until the last construction step, whose associated field extension is large enough to contain . Moreover, we obtain a sequence of fields such that each field is the quadratic extension of the previous one i.e. the fields are nested like the diagram shown below.

Iterated quadratic extensions

We can express the sequence of fields mentioned above as follows: There are the so-called iterated quadratic extensions of . Here is the main result: Any constructible number is an element of some iterated quadratic extension of . Conversely, each element of any iterated quadratic extension is a constructible number. Therefore, the set of all constructible number is the union of all iterated quadratic extensions of .