The fundamental Theorem of Algebra

This video illustrates a proof of the Fundamental Theorem of Algebra: "Every polynomial degree at least 1 with complex coefficients has a least one complex zero".

Proof: Let P(z)= z^k + a_k-1 z^k-1 + ... + a_1 z + a_0 be any polynomial (in the video P has degree 3). Substituting z=r exp(2πt) we obtain for fixed r a continuous map from the unit circle to C. For the special polynomial z^k the image of this map winds k times around the origin.

Using the deformation z^k+λ(P(z)-z^k) we can continuously deform the image of zk into the image of P(z). Now consider the map of the unit disk D to C given by P (green areas in the video). If we can prove that 0 lies inside the image of D, we have proven that P has a zero.

To do this we increase the radius r. If r is bigger than |a_k-1| + ... + |a_0| the difference between the image curves of z^k (in the video identified with the circle of radius r around the origin) and P(z) becomes smaller than the radius r (again shown in green). By the Theorem of Rouché both images must then have the same positive winding number k around 0. It follows that 0 must be inside the area the image of D (otherwise the image curve would have winding number 0). q.e.d.

This video was produced by students I.Kenig, D.Tiessen, A.Timm and V.Wittman of Hans-Christian's topology seminar 2004 at Leibniz Universität Hannover.


This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.

http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1

Category:

Howto & Style

Tags:

• topology
• fundamental
• theorem
• algebra
• zero
• proof
• path
• winding
• number
• homotopy
• bothmer

[whiutecloud24]

I think you made a mistake!

[bothmer]

What do you think is wrong? I would like to post corrections on any mistake you find.

[nihil1]

You could add voice to the explanation (and make the video a bit slower). It would make the video just perfect. 5/5 still. XD

[km2711]

Not didactic by any stretch...

Not much purpose served here...

[kykittykat]

bullshit- some need a bit of guidance jerk

[Mizbonita08]

Math is truely amazing, if we understand it, sure it may be fustrating to learn, but once you understand it ..it is amazing

[farflungfloyd]

audio, please!

[copernicus633]

These are excellent videos. However, without audio, they only convey definite meaning to those who already know the proof. In this case, we need audio which defines the "winding number" ,the role of continuity, and the dominance of the high order term.

[Rrabelo]

excelent! Hope that this type of video gets more popular. I think that is just amazing form to teach/learn/talk about math (without forget the usual ways). It is more human!

Congratulations!

[Shindashi]

Maybe, but surely you know algebraic formulations cannot exist for arbitrary polynomials; if not, Wiki "Abel." Group Theory, in particular Galois, is at the centre of the issue.

[hsdkahkjcbaksc]

I fucking hate Math !!! :D