Irreducibilité
Irreducibility is a fundamental concept in abstract algebra, particularly in the study of polynomial rings and modules. A polynomial is considered irreducible over a field if it cannot be factored into the product of two non-constant polynomials with coefficients in that same field. In simpler terms, it's a polynomial that cannot be "broken down" into simpler polynomial components within that algebraic structure. For example, the polynomial x^2 + 1 is irreducible over the field of real numbers because there are no real numbers whose squares are -1. However, over the field of complex numbers, it factors into (x - i)(x + i), making it reducible.
This concept extends to other algebraic structures. For instance, in the context of modules, an irreducible