algebra over a field

Algebra over a field involves working with mathematical objects called elements within a set that includes operations like addition, subtraction, multiplication, and division (excluding division by zero). This set, called a field, has properties ensuring these operations behave consistently—similar to rational or real numbers. Algebra over a field studies how elements combine and relate within this structure, allowing for solving equations and understanding patterns. It provides a framework that extends number systems, enabling advanced mathematical reasoning and applications in areas like geometry, physics, and computer science.