Topic of Research Seminar: Symmetry and Solvability: A Journey Through Galois Theory

Abstract: What does the tragic tale of a 19th-century genius, dying in a duel at the age of 20, have to do with the solvability of mathematical equations? Everything.

This talk recounts the story of Évariste Galois, who, on the night before his death, poured his genius onto paper to solve a puzzle that had defied the greatest minds for centuries: Why can we find general formulas for quadratic, cubic, and quartic equations, but the quintic equation remains forever out of reach? Galois’s revolutionary insight was to shift the perspective from direct calculation to the study of symmetry. He discovered that the answer lay not in manipulating the equation itself, but in studying the hidden symmetries between its solutions.

We will explore the elegant framework he developed, that is, a perfect mathematical “Rosetta Stone” known as the Fundamental Theorem of Galois Theory. This theorem establishes a breathtaking correspondence between the world of field extensions and the world of symmetry groups, providing a clear and definitive criterion for solvability by radicals. It explains, with devastating clarity, why the quintic equation is fundamentally unsolvable.

Discover how a theory born from tragedy did not just solve an ancient problem but became a cornerstone of modern mathematics, reshaping fields from number theory and geometry to cryptography.

Subject Field of Topic: Abstract Algebra

Name of Speaker: Muhammad Usman Rashid

Professorial Rank of Speaker: PhD Scholar (Mathematics Dpt.)

Email of Speaker: [email protected]

Affiliation of Speaker: School of Natural Sciences (SNS-NUST)

Date and Venue: (Wednesday) 01 October, 2025, 03:30 PM, Room # 303 Old Building, School of Natural Sciences (SNS), NUST Islamabad