Enroll Course: https://www.coursera.org/learn/theorie-de-galois
Have you ever wondered if there’s a fundamental reason why some polynomial equations can be solved using simple radicals (like square roots and cube roots), while others, like the quintic equation, famously cannot? The answer lies in the elegant and powerful framework of Galois Theory, and Coursera’s “Introduction à la théorie de Galois” offers a fantastic journey into this abstract yet profoundly impactful area of mathematics.
This course, taught in French, delves into the heart of Galois Theory, starting with the classical problem of the non-solvability of polynomial equations. It then progresses to more advanced techniques for calculating Galois groups by means of reduction modulo a prime number. The central theme is the study of polynomial roots, specifically exploring whether these roots can be expressed in terms of the polynomial’s coefficients. Evariste Galois’s genius was in recognizing the symmetries of these roots and associating a group of permutations with each polynomial.
The syllabus is meticulously structured, beginning with a gentle introduction to the problem and foundational results on single-variable polynomials. From there, it systematically builds the necessary algebraic machinery. You’ll explore field extensions, including algebraic elements and algebraically closed fields, and tackle the concept of minimal polynomials and conjugate elements. The course then ventures into finite fields, introducing Frobenius automorphisms and extensions of finite fields. Essential group theory concepts, such as Lagrange’s theorem, are covered before the pivotal Galois correspondence is explained, along with Artin’s lemma and the definition of Galois groups. The course revisits group theory to discuss solvable groups and the non-solvability of the symmetric group S_n for n >= 5. Finally, it covers cyclotomy, the solvability theorems, and the insightful method of reduction modulo p to understand Galois groups of polynomials with integer coefficients, concluding with further applications and cyclotomy over Q.
While the course is in French, the mathematical concepts are universal. For those with a solid foundation in abstract algebra (field theory and group theory basics), this course is an excellent opportunity to deepen your understanding and appreciate the beauty of Galois Theory. The progression from basic concepts to advanced applications is logical and well-paced. It’s a challenging but incredibly rewarding experience that provides a new perspective on solving equations and understanding the structure of mathematical objects.
I highly recommend “Introduction à la théorie de Galois” to any serious student of mathematics, particularly those interested in abstract algebra, number theory, or algebraic geometry. It’s a rigorous yet accessible exploration of a cornerstone of modern mathematics.
Enroll Course: https://www.coursera.org/learn/theorie-de-galois