# Math 31: Topics in Algebra

## Summer 2013

The picture on the front page of the course website is a collage of portraits of famous mathematicians, each of whom 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 he 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 born in modern-day Italy. Like many of the great mathematicians of his time, he made contributions to many different areas of mathematics, including his development of the theory of differential equations and the calculus of variations. In particular, he performed some very early work in abstract algebra and number theory, and one of the most fundamental results in elementary group theory is named in his honor. He has been described as a "timid and nervous" man, but he was highly respected during his lifetime.

• 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 lay the foundation for group theory as we know it today, and he was even the first person 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 memory of him.

• 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, among other areas. In algebra, there are several results in ring theory (specifically regarding rings of polynomials) bearing his name. He also gave the first correct proof of the Fundamental Theorem of Algebra. Modern number theory grew out of the work of Gauss, with most of his contributions contained in his 1801 book Disquisitiones Arithmeticae. In this book, he showed algebraically that a regular heptadecagon (which can be found on the cover of our textbook) can be constructed with a compass and straightedge alone.

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, which (along with the Fields Medal) is sometimes thought of as a "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 (along with William Rowan Hamilton). He was the also the first person to use the modern definition of a group, and a fundamental theorem in group theory (aptly known as Cayley's Theorem) is due to him.

• Richard Dedekind (1831–1916) Dedekind was born in modern-day Germany, and his major mathematical contributions lie in abstract algebra and the theory of real numbers. Perhaps most notably, he developed a rigorous method for constructing the set of real numbers via Dedekind cuts. In algebra, he coined the term ideal for one of the fundamental notions in ring theory. His work also heavily influenced the field of algebraic number theory, where one of the key objects of study is a particular type of ring called a Dedekind domain.