Évariste Galois

25 October 1811 – 31 May 1832

Évariste Galois, a prodigious mathematical genius, left an indelible mark on the field despite his tragically short life, ending at just 20 years old. He is most renowned for proving the impossibility of a general solution akin to the quadratic formula for polynomial equations of degree five and higher. Galois met an untimely end in a duel, a dramatic close to a life brimming with intellectual fervor. His most significant contribution lies in developing the abstract concept of a group, a foundational element in modern algebra, establishing him as a pivotal figure in the transition to more abstract mathematical thinking.
4 oil painting portrayals of how Galois may have looked based on his surviving sketch

Major Discovery

Insolubility by Radicals of Polynomial Equations of Degree 5+

Évariste Galois' major discovery was the proof of the insolubility by radicals of polynomial equations of degree five and higher. This groundbreaking work established that, unlike quadratic, cubic, and quartic equations which can be solved using algebraic formulas involving only radicals (roots), there is no general formula using radicals to solve quintic (fifth degree) and higher degree polynomial equations. Galois achieved this by linking the solvability of these equations to the properties of groups, a concept he significantly developed, which relates to the symmetries of the roots of the equation. This profound insight not only solved a longstanding mathematical problem but also laid the foundation for modern group theory and abstract algebra.

Key Facts about Évariste Galois

  • Mathematical Genius

    Galois demonstrated exceptional mathematical ability from a young age, particularly after enrolling in the Lycée Louis-le-Grand, where he was profoundly influenced by the works of mathematicians like Legendre and Lagrange.

  • Challenges with Academic Institutions

    Despite his brilliance, Galois struggled with formal education and failed twice to gain admission to the prestigious École Polytechnique.

  • Creating the Group

    Galois was indeed instrumental in the creation and conceptualization of group theory as a powerful mathematical tool. Before his work, algebra as a field was established, but the concept of a group, especially in the context of algebra, was not well-defined or systematically explored. Galois' innovative approach to solving polynomial equations laid the groundwork for group theory, which later evolved into a central structure in modern abstract algebra. His contributions were pivotal in introducing and formalizing the concept of a group, which has since become a fundamental concept in various branches of mathematics.

  • Political Involvement and Turbulence

    His life was marked by political activism and turmoil, including involvement in the revolutionary movements of his time, leading to multiple arrests and expulsions from educational institutions.

  • Tragic and Early Death

    Galois died at the age of 20 due to injuries sustained in a duel, under circumstances that remain somewhat mysterious. The night before his death, he wrote a letter outlining his mathematical ideas, which posthumously became foundational to modern algebra.

Mathematical Contributions

  • Group Theory

    Galois introduced the concept of groups in the context of algebra, providing a framework for understanding the symmetries and structural properties of algebraic systems.

  • Galois Theory

    His work laid the foundation for what is now known as Galois Theory, a branch of abstract algebra that connects field theory and group theory, and is crucial in understanding the solvability of polynomial equations.

  • Solutions by Radicals

    Galois proved that polynomial equations of degree five and higher cannot generally be solved using radicals (roots), a significant departure from the solvability of lower-degree equations like quadratic, cubic, and quartic equations.

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