The Fundamental Theorem of Algebra

  • #1
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I just wanted to say first of all that I am not looking for any specific answers, just hoping someone could shed a light on the subjects at hand.

Is the quadratic formula a specific example of some general root finding algorithm that solves for the n (or n-1?) roots of a nth degree polynomial? Kind of like the Pythagoras theorem is a specific example of the law of cosines for right triangles?

How do I understand those concepts intuitively? Should I learn complex analysis to get how the complex roots work out?
 

Answers and Replies

  • #2
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Great question.

I think that by asking these types of questions you will help yourself to understand complex concepts intuitively. So keep it up.

The quadratic formula gives us the real and complex roots of a 2nd degree polynomial. We can not use it to solve higher degree polynomials.

Sometimes the quadratic will just give us one answer for x. This is a case where we have found a root with multiplicity 2. This concept is a bit tough to explain. Here is a link to give an exact definition. See if you can understand it: Wikipedia.

Complex analysis will certainly help, but I think the real meat and potatoes is in a general field of mathematics that some people refer to as Abstract Algebra.

I hope I have helped.
 
  • #3
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Thanks! Would you say I should learn complex analysis first or abstract algebra? Or are the two unrelated enough that it won't matter?
 
  • #4
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Thanks! Would you say I should learn complex analysis first or abstract algebra? Or are the two unrelated enough that it won't matter?
The two have small intersection at the introductory level. Neither are prerequisites for the other. But I have found that there are junior level complex analysis classes, while algebra does not come till senior level.
 
  • #5
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The two have small intersection at the introductory level. Neither are prerequisites for the other. But I have found that there are junior level complex analysis classes, while algebra does not come till senior level.
That is a great observation! There definitely is a lot of intersection between the two fields.

And V0ODO0CH1LD, I would talk to your teachers and/or professors and see what they think. But personally when I was in school, I took an Abstract Algebra introductory class with just a basic knowledge of complex numbers. I'd say you could take an first course in either, in any order and turn out OK.
 
  • #6
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That is a great observation! There definitely is a lot of intersection between the two fields.

And V0ODO0CH1LD, I would talk to your teachers and/or professors and see what they think. But personally when I was in school, I took an Abstract Algebra introductory class with just a basic knowledge of complex numbers. I'd say you could take an first course in either, in any order and turn out OK.
Right, since complex analysis is not considered on the main track, some people take algebra without ever having taken complex analysis. There is some flexibility in paths through a math department.

For instance, a computer scientist might conceivably take algebra (maybe combinatorics is even more likely) but not complex analysis, while the the closely related field of electrical engineering might use more from complex analysis.
 
  • #7
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7
I just wanted to say first of all that I am not looking for any specific answers, just hoping someone could shed a light on the subjects at hand.

Is the quadratic formula a specific example of some general root finding algorithm that solves for the n (or n-1?) roots of a nth degree polynomial? Kind of like the Pythagoras theorem is a specific example of the law of cosines for right triangles?

How do I understand those concepts intuitively? Should I learn complex analysis to get how the complex roots work out?
There's a formula for 2nd degree equations -- quadratics. There's a formula for third-degree equations (cubics). There's a formula for fourth degree equations (quartics). But there is NO general formula for solving 5th degree equations or higher.

To learn about that, you'd study abstract algebra.

What you'd learn from complex analysis is a proof that every polynomial of degree n has n complex roots. But there are other proofs that don't explicitly use complex analysis. But from complex analysis you'd understand why the n roots of the polynomial zn = 1 are the vertices of a regular n-gon in the plane -- a very cool fact indeed.

So it sounds like you'd be more interested in abstract algebra if you want to learn about formulas for finding roots; and complex analysis if you're interested in the roots in general.

Here's a fascinating article about the beautiful images you get when you plot all the roots of various classes of polynomials.

http://math.ucr.edu/home/baez/roots/
 
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