## 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

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

## Women’s Week #1: Sofya Kovalevskaya

This weeks theme is Women, thus I will be celebrating the stories of three female mathematicians. Hope you enjoy them!

Sofya Kovalevskaya, born in Russia in 1850, was a mathematician and writer who made important contributions to partial differential equations. She was the first woman appointed to a full professorship in Northern Europe and was also one of the first women to work as the editor of a scientific journal.

Kovalevskaya was the middle child of two well-educated members of the Russian nobility. Whilst being educated by tutors and governesses, Kovalevskaya was attracted to mathematics in particular; Sofya wrote in her autobiography:

“The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.”

“I began to feel an attraction for my mathematics so intense that I started to neglect my other studies.”

Sofya was forced to marry so that she could go abroad for higher education, as women in Russia could not live apart from their families without written permission of their father or husband and her father would not allow her to leave home to study at a university. Thus, she entered a marriage of convenience to Vladimir Kovalevski, who was a palaeontologist.

In 1869, Sofya travelled to Heidelberg to study mathematics and natural sciences. When she discovered that women could not matriculate at the university, she persuaded the university authorities to allow her to attend lectures unofficially – a similar situation to that of Emmy Noether.

Kovalevskaya moved to Berlin in 1871 to study with Weierstrass (Leo Königsberger’s teacher).

“These studies had the deepest possible influence on my entire career in mathematics. They determined finally and irrevocably the direction I was to follow in my later scientific work: all my work has been done precisely in the spirit of Weierstrass”

By 1874, she had completed three papers, which Weierstrass deemed worthy of a doctorate. The three papers were on:

• Partial differential equations: contains what is now commonly known as the Cauchy-Kovalevskaya theorem, which gives conditions for the existence of solutions to a certain class of those equations. It was published in Crelle’s journal, which was an incredible honour for an unknown mathematician.
• Abelian integrals: concerned with the reduction of abelian integrals to simpler elliptic integrals.
• Dynamics of Saturn’s Ring.

In 1883, Vladimir committed suicide. After the initial shock, Kovalevskaya immersed herself deeper into her mathematical work. Due to Mittag-Leffler‘s help, Sofya obtained a position as privat docent in Stockholm. After she began to lecture there, she was quickly appointed to a five year extraordinary professorship, becoming the first women since Laura Bassi and Maria Gaetana Agnesis to hold a chair at a European university.

During her time in Stockholm, she is considered to have done her most important research in the field of analysis. Additionally, she became an editor of the journal Acta Mathematica.

In 1886, she was awarded the Prix Bordin for her paper that contains the discovery of the ‘Kovalevskaya Top‘, which is the only other case of rigid body motion that is completely integrable (besides the tops of Euler and Lagrange).

Kovalevskaya’s last published work was a short article, which gave a new, simpler proof of Bruns’ theorem on a property of the potential function of a homogeneous body. In early 1891, at the height of her mathematical reputation, Kovalevskaya died of influenza complicated by pneumonia.

### Tributes

• Sonya Kovalevsky High School Mathematics Day: a grant-making program of the Association for Women in Mathematics (AWM) to fund workshops across the US, which encourage girls to explore mathematics.
• Sonya Kovalevsky Lecture: sponsored annually by the AWM, it highlights significant contributions of women in the fields of applied or computational mathematics.
• The lunar crater Kovalevskaya is named after her.

Sources: 1 | 2 | 3

Stay tuned for Wednesday’s post! M x