Hi,

I am an engineer working with some algorithms that uses a lot of abstract algebra. And as I explain below, this is sometimes a bit difficult for me to understand.

The abstract algebra involved are: differential fields, and field extensions.

The extensions fields are simple transcendental differential extensions to the field k. If K denotes such an extension field, t is said to be in K and a monomial over k, and Dt is a polynomial in k[t] -- where D is the derivation in the field.

The book, that I am reading defines some sets of polynomials:

$\displaystyle

S_{k[t]:k} = \{ p \in k[t] \text{ such that p is special w.r.t D}\}

$

$\displaystyle

S_{k[t]:k}^{irr} = \{ p \in S_{k[t]:k} \text{ such that p is monic and irreducible}\}

$

where special means: gcd(p,Dp)= p.

And

$\displaystyle

S_{1,k[t]:k} = \{ p \in k[t] \text{ such that p is special of the first kind}\}

$

$\displaystyle

S_{1,k[t]:k}^{irr} = \{ p \in S_{1,k[t]:k} \text{ such that p is monic and irreducible}\}

$

Note, that it is necessary to know the precise meaning of "special of first kind". It is just some advanced math related to a polynomial in k[t]. Note also, that when the monomial extension is clear from the context, the subscripts are omitted.

My problem with these definitions are, that I do not always understand how to interprete them, when they occur in the pseudocode for the various algorithms listed in the book.

Often the following definition is used:

$\displaystyle

p \in S^{irr}

$

Which is easy to understand.

In other cases, the definitions are used quite different. For Instance:

$\displaystyle

\text{if } S_1^{irr} = S^{irr} \text{ then ...}

$

This, I find difficult to understand, since I cannot figure out how to relate the definitions (sets) to a specific polynomial, or other elements in the context.

If the two sets are equal, what makes them equal?, and how are their content defined?

Hope someone can help me a bit on this.

Kind regards,

ASL