Why is the formal derivative of a polynomial important in algebra?

[tgt]

Do people agree with this?

 

[cristo]

In what sense?

 

[alxm]

I've often considered algebra to be similar to a flight data recorder.

 

[Phrak]

I've often considered algebra to be similar to a flight data recorder.

Ahh, the orange colored black box.

 

[HallsofIvy]

Am I the only one who is completely puzzled by this?

 

[confinement]

I relate to this comment in the sense that I have often felt that results in algebra, or obtained in other fields using algebra, are the most powerful and yet the most unexpected and difficult to discover or prove.

 

[MathematicalPhysicist]

Am I the only one who is completely puzzled by this?

It's a post modern post, just flow with it. (-:

 

[HallsofIvy]

I think the original question could be paraphrased as "how many of you don't understand algebra"!

 

[Klockan3]

Algebra is not like a black box, algebra is like a shining beacon of logic.

 

[csprof2000]

Algebra is just like a black box, although not black, and not boxy.

 

[tgt]

I think the original question could be paraphrased as "how many of you don't understand algebra"!

Consider the map D:R[x]->R[x] where R is a ring. A polynomial p in R[x] is mapped to dp/dx (differentiation of the polynomial) in R[x]. This map is just following some 'blind' rule. It works but it ignores the machinery of differentiation which is that of analysis. The algebraist is just following 'black box' instructions in this case.

Note that I use to be a huge fan of algebra but the more I learn about it, the more the black box nature of it come to me.

 

[farleyknight]

Note that I use to be a huge fan of algebra but the more I learn about it, the more the black box nature of it come to me.

Coming from a comp-sci background, I agree with this.. Functions from a comp-sci background are from a bottom up perspective: Here's the way the function works, what do we know about it?.. Algebra, on the other hand, seems to be looking from the top down: Here's the rules the function obeys, what do we know about it?

Given this, I'm actually quite surprised that the comp-sci curriculum at most universities don't require more abstract algebra. Category theory at least.. As far as I know, there aren't a lot of direct correspondences between algebra and system specifications, but so far they seem to be very similar in spirit.

 

[Hurkyl]

Functions from a comp-sci background are from a bottom up perspective: Here's the way the function works, what do we know about it?..

Not really; you need the other direction too. The user of a library, for example, only cares about "the rules the function obeys", and good library design works hard to abstract away the details the user doesn't need to worry about. One of the main topics in computability theory is, given the "rules the function obeys", what sorts of execution models (if any) could compute it?

 

[matt grime]

works but it ignores the machinery of differentiation which is that of analysis. The algebraist is just following 'black box' instructions in this case.

What do you mean by 'it just works'? How is it ignoring the machinery of differentiation? Those just seem like slogans without justification as to why algebra ought to pay attention to analysis, and why you can demonstrate that it hasn't.

Actually, differentiation can be given an algebraic treatment: it is called the ring of dual numbers, or

R[e]/e^2

and this allows one to generalize differentiation to places where analysis doesn't make sense.There are also some nice ways to characterise integration and differentiation as the unique linear operators satisfying some rules, though I forget which: it was a STEP question in the mid 90s.

 

[djeitnstine]

Algebra has to be one of my least favorite but most useful tools D= in many cases it is my first choice of last resorts. =]

 

[MathematicalPhysicist]

Given this, I'm actually quite surprised that the comp-sci curriculum at most universities don't require more abstract algebra. Category theory at least.. As far as I know, there aren't a lot of direct correspondences between algebra and system specifications, but so far they seem to be very similar in spirit.

Well, there's a new (relatively) textbook on this called:
"category theory for software engineering"
perhaps it can be a good read for you.

I myself don't find the need to take a course in algebra (besides the courses in Linear Algebra which were mandatory anyway), I've got MacLane's textbook which seems quite complete, and if I need other textbooks in algebra I know where to find them. (-:

 

[matt grime]

The rules of algebra aren't just pulled from a hat, any more than any other mathematical constructs. As far as I can tell, it seems that you want some kind of physical meaning behind them.

You might wonder why differentiation is used at all in the study of polynomial rings: as you say what on Earth does the notion of slope mean?

There is a large body of work generalizing geometry, say, to positive characteristic, but that is too obtuse to be a satisfying answer.

Instead, let us put it this way: just because one can define the formal derivative of a polynomial over any field, why would you want to do it? Surely there are more natural linear maps from F[x] to itself, like L:x^n --> x^{n-1}, or R the map in the direction. Note that LR=Id, but RL=/=Id, and that's a nice result, and an important one.

So, why the formal derivative? Why do we even mention that it can be applied to polynomial algebras? How about this result:

Let g be a polynomial in F[x], then if the hcf of g and g' is 1, g has no repeated factors.

It is not as though algebra and analysis grew up in isolation from one another. Different subjects borrow ideas from each other all the time. The original motivation for differentiation might have been to find rates of change, but one rapidly realizes that its purely formal properties, such as the chain rule, product rule, can be generalized to other places.

Incidentally, you should try to prove what the Lie algebra of SL(2,R) is without using dual numbers.