What is field extension?

A field extension is a mathematical concept that describes the expansion of a given field by adding new elements to it. A field is a set of numbers that follows a specific set of rules for addition, subtraction, multiplication, and division. Extensions expand the set of numbers that can be used for calculations within a given field. In other words, a field extension is a larger field that includes the original field as a subset.

Field extensions can be either algebraic or transcendental. An algebraic field extension is one in which the added elements are roots of polynomials with coefficients in the original field. A transcendental field extension is one in which the added elements are not roots of any such polynomial. One of the most common examples of an algebraic field extension is the extension of the rationals by the square root of 2.

Field extensions play an important role in algebraic geometry, number theory, and other areas of pure mathematics. They also have practical applications in cryptography, coding theory, and other areas of computer science.