F.T.A. via Complex Analysis

Although this requires a bit of knowledge on Complex Anlaysis, I recently discovered this new way to prove the Fundamental Theorem of Algebra and I couldn’t help but share it.

First of all, what is the Fundamental Theorem of Algebra (FTA)? This very important (hence the name!) result states that:

Every non-constant polynomial with complex coefficients has a complex root. 

In order to prove this, we must first be aware of Liouville’s Theorem:

Every bounded, entire function is constant. 


Definitions

Bounded: a function on a set X is said to be bounded if there exists a real number M such that

|f(x)|\le M

for all x in X.

Entire: An entire function is a holomorphic function on the entire complex plane.


Liouville’s theorem is proved using the Cauchy integral formula for a disc, one of the most important results in Complex Analysis. Although I will not describe how to prove it or what it states in this blog post, I encourage you to read about here it as it is truly a remarkable result.

Now armed with Liouville’s Theorem we can prove the FTA.

Proof

Let P(z) = zn + cn-1zn-1 + … + c1z + c0 be a polynomial of degree n > 0. Then |P(z)| –> ∞ as |z| –> ∞, so there exists R such that |P(z)| > 1 for all z with |z| > R.

Consider f(z) = 1/P(z). If P has no complex zeros then f is entire. So, as f is continuous, f is bounded on {|z| ≤ R}.

As |f(z)| < 1 when |z| > R, f is a bounded entire function, so by Liouville’s Theorem f is constant, which is a contradiction.

The only thing we assumed was that P had no complex zeros, and so we contradicted this fact. Hence, P must have at least one complex zero. Amazing right!

 

M x

 

Diophantine Approximation: Liouville’s Theorem

Diophantine approximation deals with the approximation of real numbers by rational numbers.

Liouville’s Theorem

In the 1840’s Liouville obtained the first lower bound for the approximation of algebraic numbers:

Let α ∈ R be an irrational algebraic number satisfying f(α) = 0 with non-zero irreducible (cannot be reduced) f ∈ Z[x] of degree d. Then there is a non-zero constant C such that for every fraction p/q

Screen Shot 2016-12-11 at 10.35.05 AM.png

Proof

The proof utilises the mean value theorem. By this theorem, given p/q, there is a real ξ between α and p/q such that

Screen Shot 2016-12-11 at 10.35.08 AM.png

Since f has integer coefficients and is of degree d, the value of f(p/q) is a rational number with denominator at worst q^d. Since f is irreducible, f(p/q) cannot be equal to 0. Thus

Screen Shot 2016-12-11 at 10.40.22 AM.png

and so

Screen Shot 2016-12-11 at 10.40.58 AM.png

A corollary of this result is that numbers that are well approximable by rational numbers, i.e. in for every d ≥ 1 and positive constant C, there is a rational p/q such that

Screen Shot 2016-12-11 at 10.43.32 AM.png

are transcendental.

Example

Letscreen-shot-2016-12-11-at-10-45-22-am

β is a real, transcendental number.

This is because there is a rational approximation

Screen Shot 2016-12-11 at 10.46.43 AM.png

with

screen-shot-2016-12-11-at-10-47-21-am

Analysing this inequality, the ratio

screen-shot-2016-12-11-at-10-48-30-am

is unbounded as N → +∞, and so β is well approximable by rationals.

M x