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Field (Mathematics): Theories
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While physics deals with an extremely wide variety of systems, there are certain theories that are used by physics as a whole, and not by any single field. Each of these theories is believed to be basically correct, within a certain domain of validity. For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at much less than the speed of light. These theories continue to be areas of active research; for instance, a remarkable aspect of classical mechanics known as chaos was discovered in the 20th century, three centuries after its formulation by Isaac Newton. However, few physicists expect any of them to prove fundamentally misguided. Therefore, they are used as the basis for research into more specialized topics, and any contemporary student of physics, regardless of his or her specialization, is generally expected to be well-versed in all of them.
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The field of mathematics today offers virtually unlimited opportunities for application, together with the aesthetic appeal and intellectual challenge that have always been so compelling to so many. It provides both the language in which the theories of many disciplines, such as physics and economics, are best expressed, and the techniques by which many problems of these disciplines are analyzed and solved. Mathematical training develops analytical skills that are of great value in understanding and resolving issues in almost any field. This is reflected in the large number of careers for which mathematical training is considered to be particularly desirable preparation.
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Electrical engineering and theoretical physics use the field of complex numbers to study electric circuits, electricity, magnetism, and quantum theory. In coding theory, fields are used to reduce the errors that arise in transmitting information over media such as telephone lines (see Information Theory). The mathematical theory of fields is one of the principal tools used to study the deeper properties of numbers.
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The concept of a field is of use, for example, in defining vector s and matrices , two structures in linear algebra whose components can be elements of an arbitrary field. Galois theory studies the symmetry of equations by investigating the ways in which fields can be contained in each other. See field theory for more information.
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An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and is used in all sciences. Numerical analysis investigates the methods of efficiently solving various mathematical problems numerically on computers and takes rounding errors into account.
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The requirement 0 ≠ 1 ensures that the set which only contains a single element is not a field. Directly from the axioms, one may show that (F, +) and (F \ {0}, *) are commutative groups (abelian groups) and that therefore (see elementary group theory) the additive inverse −a and the multiplicative inverse a−1 are uniquely determined by a. Other useful rules include
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