Field: In mathematics

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers, and the field of complex numbers. Many other fields, such as fields of rational functions ... In abstract algebra, a field is a type of commutative ring in which every nonzero element has a multiplicative inverse; in other words, a ring \ (F\) is a field if and only if there exists an element \ (e\) such that for every \ (a \in F\), there exists an element \ (a^ {-1} \in F\) such that \ [a \cdot a^ {-1} = a^ {-1} \cdot a = e.\] Fields are useful in forming the scalars in a given vector space; for instance, polynomials can draw coefficients from the field of real numbers, but also may ... Learn the meaning and usage of the word field in different contexts, such as grassland, sports, mining, computing, and more. Find synonyms, related words, and pronunciation guides for field . Spelling of Field : Field is spelled f-i-e-l-d . Definition of Field : A field is an open area of land free of woods and buildings. There are a variety of types of fields, each dedicated to different activities or purposes. Field may also refer to the area or division of an activity, subject, or profession, or to the location in which a specific type of work is completed for a particular industry. A field is also a region or space in which a particular effect exists, such as a magnetic field ...