Marie-Sophie Germain ( 1 April 1776 – 27 June 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library, including ones by Euler, and from correspondence under the pseudonym of Monsieur Le Blanc with famous mathematicians, such as Lagrange, Legendre, and Gauss. One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject. Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after. Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life.
Family
Marie-Sophie Germain was born in a house on Rue Saint-Denis on 1 April 1776, in Paris, France, to Marie-Madeline (née Gruguelin) and Ambroise-François Germain and was baptised the same day. She was from a bourgeois family. According to most sources, her father was a wealthy silk merchant, though some believe he was a goldsmith. In 1789, he was elected as a representative of the bourgeoisie to the États-Généraux, which he saw change into the National Assembly. It has been assumed that Sophie witnessed many discussions between her father and his friends on politics and philosophy. Gray proposes that after his political career, Ambroise-François became the director of a bank. The family remained well-off enough to support Germain throughout her adult life.
Marie-Sophie had one older sister, Marie-Madeline and one younger sister, Angélique-Ambroise. Her mother was also named Marie-Madeline, and this plethora of "Maries" may have been the reason she went by Sophie. Germain's nephew Armand-Jacques Lherbette, her sister Marie-Madeline's son, published some of Germain's work after she died.
Introduction to mathematics
When Germain was 13, the Bastille fell, and the revolutionary atmosphere of the city forced her to stay inside. For entertainment, she turned to her father's library. Here she found Jean-Étienne Montucla's L'Histoire des Mathématiques, and his story of the death of Archimedes intrigued her. Germain thought that if the geometry method, which at that time referred to all of pure mathematics, could hold such fascination for Archimedes, it was a subject worthy of study. So she pored over every book on mathematics in her father's library, even teaching herself Latin and Greek, so she could read works like those of Isaac Newton and Leonhard Euler. She also enjoyed Traité d'Arithmétique by Étienne Bézout and Le Calcul Différentiel by Jacques Antoine-Joseph Cousin. Later, Cousin visited Germain at home, encouraging her in her studies.
Later work in elasticity
Germain published her prize-winning essay at her own expense in 1821, mostly because she wanted to present her work in opposition to that of Poisson. In the essay she pointed out some of the errors in his method. In 1826 she submitted a revised version of her 1821 essay to the academy. According to Andrea Del Centina, the revision included attempts to clarify her work by "introducing certain simplifying hypotheses". This put the academy in an awkward position, as they felt the paper to be "inadequate and trivial", but they did not want to "treat her as a professional colleague, as they would any man, by simply rejecting the work". So Augustin-Louis Cauchy, who had been appointed to review her work, recommended her to publish it, and she followed his advice.
One further work of Germain's on elasticity was published the year of her death, posthumously, in 1831, her "Mémoire sur la courbure des surfaces". She used the mean curvature in her research.
Her work on Fermat's Last Theorem
Fermat's Last Theorem can be divided into two cases. Case 1 involves all powers p that do not divide any of x, y, or z. Case 2 includes all p that divide at least one of x, y, or z. Germain proposed the following, commonly called "Sophie Germain's theorem" Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes p < 100, but according to Andrea Del Centina, "she had actually shown that it holds for every exponent p < 197". L. E. Dickson later used Germain's theorem to prove the first case of Fermat's Last Theorem for all odd primes less than 1700.