Constructing Subalgebras of K[x] Using the Minimal Polynomial

Abstract: In this report we will be working with subalgebras A of finite codimension in K[x]. It is known that such subalgebras can be expressed using a set of linear conditions evaluated at a finite set of points called the spectrum elements of A. These conditions are of one of two types, equality conditions or α-derivations, which in turn consists of the values and the values of the derivations of the elements in our algebra. From this representation we find a way to construct a polynomial, the zeros of which are exactly the spectrum element. This polynomial, called the minimal polynomial of A, has the property that its product with an arbitrary polynomial lies in our algebra. In order to find subalgebras of A we can add an additional condition, namely an α-derivation, where α lies in the spectrum of A. To find all such α-derivations, which can be written as a linear combination of regular derivations, we find an upper limit to the order of the derivations involved. To fully determine the derivation we also construct a method of finding all the required restrictions on the coefficients of said linear combination.