# Polynomial proof that nth degree poly has n roots

• Dec 12th 2012, 01:34 AM
mathlover10
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?
• Dec 12th 2012, 12:14 PM
Drexel28
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.
• Jan 1st 2013, 02:02 AM
mathlover10
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)
• Jun 17th 2013, 09:21 PM
Drexel28
Re: Polynomial proof that nth degree poly has n roots
Quote:

Originally Posted by mathlover10
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
• Jun 19th 2013, 07:02 PM
mathlover10
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