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Math Help - Polynomial proof that nth degree poly has n roots

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    Polynomial proof that nth degree poly has n roots

    where can I please find the proof of the fundamental result of algebra that says that a polynomial of degree n has n real/complex roots? is that covered in most algebra texts?

    I have read some of the solution methods that use factorization, inspection to find roots of say n=3, 4 polynomials, also partial differentiation, are there others?
    Last edited by mathlover10; December 12th 2012 at 12:38 AM.
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    Re: Polynomial proof that nth degree poly has n roots

    There are tons and tons of proofs of this fact. Probably the quickest I know is that the non-constant polynomial p is a holomorphic map \mathbb{C}\to\mathbb{C} and by setting p(\infty)=\infty induces a holomorphic map between the Riemann sphere and itself S^2\to S^2. But, it is a common fact that a holomorphic map between Riemann surfaces where the domain is compact, must be surjective, and so p is surjective. But, since p(\infty)=\infty we see that there must exist some z\in\mathbb{C} such that p(z)=0. Thus, you have that p has one zero, and the fact that it has n comes from basic algebra.
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    Re: Polynomial proof that nth degree poly has n roots

    Thanks very much for your beautiful and elegant proof with the wonderful Gaussian Sphere!!! So Galois theory is one of the approaches to this problem...
    (So sorry for the delay in replying)
    Last edited by mathlover10; January 1st 2013 at 01:10 AM.
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    MHF Contributor Drexel28's Avatar
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    Re: Polynomial proof that nth degree poly has n roots

    Quote Originally Posted by mathlover10 View Post
    Thanks very much for your beautiful and elegant proof with the wonderful Gaussian Sphere!!! So Galois theory is one of the approaches to this problem...
    (So sorry for the delay in replying)
    Yes, Galois Theory is one approach to show this. It basically comes down to using the fact that every polynomial of odd degree is irreducible over [\tex]\mathbb{R}[/tex], and a bit of Sylow theory.

    For a vast generalization, I suggest you take a look at the Artin-Schreier Theorem.

    Best,
    Alex
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    Re: Polynomial proof that nth degree poly has n roots

    Thanks very much!!!!!!!!!!!!!!!!!!!!!!!!!!!!! it's interesting I came across an article where a result in harmonic extensions of the fundamental theorem of algebra solved a conjecture from gravitational lensing research http://www.ams.org/notices/200806/tx080600666p.pdf
    Last edited by mathlover10; June 19th 2013 at 06:05 PM.
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