# Thread: Problems with abstract algebra (differential fields) in algorithms

1. ## Problems with abstract algebra (differential fields) in algorithms

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:

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

$
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

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

$
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:

$
p \in S^{irr}
$

Which is easy to understand.

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

$
\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

2. Maybe the choice of extension field with differential can make the two sets equal? It would be easier to help if you gave more context.

3. ## Re: Problems with abstract algebra (differential fields) in algorithms

The example I give above:

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

occurs in an algorithm that takes as input: the derivation D, and some functions f, and w1,w2,.... in k(t).

The text explaining the algorithm writes: this gives an algorithm that reduces the problem to one over k(t) if $S_1^{irr} = S^{irr}$

And you may be right.

It is probably the monomial t in the extension field which makes the sets equal. Or more precise: Dt which is a polynomial in k[t].