The notation k[X] represents the set of all polynomials with coefficients from a field k. In this context, a polynomial is an expression formed by combining variables raised to non-negative integer powers, multiplied by coefficients from k. For example, in k[X], a polynomial might look like a_0 + a_1X + a_2X^2, where a_0, a_1, a_2 are elements of the field k. The structure k[X] is important in algebra because it forms a ring, meaning it supports addition and multiplication of polynomials while maintaining certain properties. This ring is fundamental in various areas of mathematics, including algebraic geometry and number theory, as it allows for the study of polynomial equations and their solutions.