Galois Representations

Galois representations are mathematical tools that connect number theory and symmetry. They describe how the symmetries of algebraic equations, especially those involving roots of numbers, act on solutions in different contexts. Essentially, they encode how the structure of solutions changes when you permute or transform the underlying numbers. By translating these symmetries into matrix forms, Galois representations enable mathematicians to study complex number systems and their properties using the language of linear algebra, providing deep insights into problems like solving equations and understanding number fields.