**Field Theory In Mathematics**: this diagram is one of our most searched charts and infographics by people seeking to learn about new things and improve their general knowledge of how the world works.

Field theory is a branch of mathematics that studies the properties of fields, which are sets with two operations: addition and multiplication. Fields are commutative rings in which every nonzero element has a multiplicative inverse. The non-zero elements of a field form an abelian group under multiplication. Field theory is a fundamental algebraic structure that is widely used in algebra, number theory, and many other areas of mathematics .

The study of fields is motivated by the desire to extend the properties of the rational and real numbers to other mathematical objects. Fields are used to define vector spaces, which are fundamental objects in linear algebra. They are also used to define algebraic extensions, which are used to study algebraic equations. The theory of fields is also used in algebraic geometry, where it is used to study algebraic varieties .

The study of fields is divided into two main branches: algebraic field theory and transcendental field theory. Algebraic field theory is concerned with the study of algebraic extensions of fields, while transcendental field theory is concerned with the study of transcendental extensions of fields. The theory of algebraic extensions is closely related to the theory of Galois groups, which are groups that describe the symmetries of algebraic equations .

One of the most important results in field theory is the fundamental theorem of algebra, which states that every non-constant polynomial with complex coefficients has a complex root. This theorem is a consequence of the fact that the complex numbers form an algebraically closed field. Another important result is the theorem of the primitive element, which states that every finite separable extension of a field is generated by a single element .

Field theory has many applications in other areas of mathematics. For example, it is used in algebraic number theory to study the properties of number fields, which are algebraic extensions of the rational numbers. It is also used in algebraic geometry to study algebraic varieties, which are geometric objects defined by polynomial equations. In physics, field theory is used to describe the behavior of physical fields, such as the electromagnetic field .

In conclusion, field theory is a fundamental branch of mathematics that studies the properties of fields, which are sets with two operations: addition and multiplication. Fields are used to define vector spaces, algebraic extensions, and algebraic varieties. Field theory has many applications in other areas of mathematics and in physics..