# Math 31: Topics in Algebra

Last updated June 6, 2012

The picture on the front page of the course website is a collage of portraits of famous mathematicians who made significant contributions to the field that we now know as abstract algebra. Some of them will likely be mentioned in class when we cover results to which they contributed. Below are descriptions of these people and their work.

Top row, from left to right:

• Niccolò Fontana Tartaglia (1499/1500 - 1557) Tartaglia was an Italian mathematician. The name "Tartaglia" is actually a nickname meaning "stammerer", a reference to his injury-induced speech impediment. He was largely self-taught, and was the first person to translate Euclid's Elements into a modern European language. He is best remembered for his contributions to algebra, namely his discovery of a formula for the solutions to a cubic equation. Such a formula was also found by Gerolamo Cardano at roughly the same time, and the modern formula is known as the Cardano-Tartaglia formula. Cardano also found a solution to the general quartic equation.

• Joseph-Louis Lagrange (1736 - 1813) Despite his French-sounding name, Lagrange was an Italian mathematician. Like many of the great mathematicians of his time, he made contributions to many different areas of mathematics. In particular, he did some early work in abstract algebra. We will learn about Lagrange's Theorem fairly soon, which is one of the most fundamental results in group theory.

• Évariste Galois (1811 - 1832) Galois was a very gifted young French mathematician, and his story is one of the most tragic in the history of mathematics. He was killed at the age of 20 in a duel that is still veiled in mystery. Before that, he made huge contributions to abstract algebra. He helped to found group theory as we know it today, and he was the first to use the term "group". Perhaps most importantly, he proved that it is impossible to solve a fifth-degree polynomial (or a polynomial of any higher degree) using radicals by studying permutation groups associated to polynomials. This area of algebra is still important today, and it is known as Galois theory in his honor.

• Carl Friedrich Gauss (1777 - 1855) Along with Leonhard Euler, Gauss is considered to be one of the greatest and most prolific mathematicians of all time. He made significant contributions to algebra, number theory, geometry, and physics, just to name a few areas. In algebra, there are several results in ring theory (specifically regarding rings of polynomials) bearing his name.

Bottom row, left to right:
• Niels Henrik Abel (1802 - 1829) Abel was a Norwegian mathematician who, like Galois, did seminal work in algebra before dying at a very young age. Strangely enough, he proved similar results regarding the insolvability of the quintic independently from Galois. In honor of his work in group theory, abelian groups are named after him. The Abel Prize in mathematics, sometimes thought of as the "Nobel Prize in Mathematics," is also named for him.

• Emmy Noether (1882 - 1935) Noether is widely considered to be the greatest female mathematician of all time, and in fact one of the greatest mathematicians ever. Her most important work was related to abstract algebra, specifically the theory of rings and fields. The concept of a Noetherian ring, as well as several theorems in algebra, are named in her honor. She became a lecturer at the University of Göttingen in 1915, at the invitation of David Hilbert. She was forced to leave in 1933, when Adolf Hitler expelled Jewish faculty members from Göttingen. She emigrated to the United States, where she took up a position at Bryn Mawr, which she held until her death in 1935.

• Arthur Cayley (1821 - 1895) Cayley was a British mathematician whose work is known to students of abstract algebra and linear algebra. The Cayley-Hamilton Theorem for matrices is named after him and William Rowan Hamilton, and a fundamental theorem in group theory, Cayley's Theorem, is due to him.

• Camille Jordan (1838 - 1922) Like Cayley, Jordan made contributions to both abstract algebra and linear algebra. He is known for developing the Jordan normal form of a matrix, and for originating the Jordan-Hölder Theorem in group theory.