Human: Can you list some common theories and speculations surrounding Évariste Galois' untimely death?
Assistant: Évariste Galois was a French mathematician who lived in the 19th century and made significant contributions to the field of abstract algebra. His death at the age of 20 is still shrouded in mystery, and several theories and speculations surround it. Here are some of the most common ones:

Duel: One of the most popular theories is that Galois died in a duel over a love affair. He was known to be involved with a woman named Stephanie-Felice Ducrest de Saint-Aubin, and it is believed that he challenged her ex-lover to a duel. Galois was shot and fatally wounded in the duel.

Political assassination: Galois was also politically active and a member of a radical Republican group. Some historians speculate that he was assassinated by the government because of his involvement in revolutionary activities.

Suicide: Another theory is that Galois was depressed and committed suicide. He had just been expelled from school, and his mathematical work had not yet been recognized. He may have been despondent about his future prospects.

Accidental shooting: There is also a theory that Galois' death was accidental. He was known to carry a gun, and it is possible that he accidentally shot himself while handling it.

Poisoning: Some researchers believe that Galois was poisoned by someone who wanted to prevent him from sharing his mathematical ideas or political beliefs.

It is important to note that these theories are largely speculative, and the exact circumstances of Galois' death remain unknown.

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Human: Perfect. Could you elaborate about his mathematical ideas ? More precisely, are any of his works still used as of today ?
Assistant: Definitely! Évariste Galois is best known for his work in the field of abstract algebra, particularly in the study of group theory. He developed a theory that laid the foundations for modern algebra, including the concept of a group, which is a mathematical structure that captures the symmetry of objects and shapes.

Galois' most famous work is his theory of Galois theory, which explores the relationship between the roots of a polynomial equation and the structure of its corresponding Galois group. This theory has wide-ranging applications in fields such as cryptography, coding theory, and number theory.

In addition to Galois theory, Galois made significant contributions to the theory of equations, number theory, and the study of elliptic functions. He also introduced the idea of solvability by radicals, which is the concept that some polynomial equations can be solved using only arithmetic operations and the extraction of roots.

Many of Galois' ideas have had a lasting impact on mathematics and are still studied and used today. His theory of Galois theory is a fundamental part of modern algebra and is widely used in fields such as algebraic geometry, number theory, and mathematical physics. The concept of solvability by radicals is also important in algebraic geometry and has applications in cryptography.

In recognition of his contributions to mathematics, Galois has been honored by having several mathematical concepts and structures named after him, including the Galois group, Galois connection, and Galois cohomology, among others.