Infinite extensions refer to the concept in mathematics where a set or structure can be expanded indefinitely. This idea is often explored in fields like algebra and geometry, where one can add more elements or dimensions without limit. For example, the set of real numbers can be extended infinitely in both positive and negative directions. In field theory, infinite extensions occur when a field is expanded by adding new elements, creating a larger field that contains the original. These extensions can be useful for solving equations that cannot be addressed within the original field, allowing for a deeper understanding of mathematical relationships.