Non User
Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z	Business Non User Industries Finance Tax

Home > Field (mathematics)

Fields are important objects of study in algebra since they provide the proper generalization of number domains, such as the sets of rational numbers, real numbers, or

The concept of a field is of use, for example, in defining vectors 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 (mathematics) for more.

1 Definition

A field is a commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, and if the multiplication operation is written as then a ' (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse.

Spelled out, this means that the following hold:

Closure of F under + and *

For all a,b belonging to F, both a + b and a * b belong to F (or more formally, + and * are binary operations on F);

Both + and * are associative

For all a,b,c in F, a + (b + c) = (a + b) + c and a * (b * c) = (a * b) * c.

Both + and * are commutative

For all a,b belonging to F, a + b = b + a and a * b = b * a.

The operation * is distributive over the operation +

For all a,b,c, belonging to F, a * (b + c) = (a * b) + (a * c).

Existence of an additive identity

There exists an element 0 in F, such that for all a belonging to F, a + 0 = a.

Existence of a multiplicative identity

There exists an element 1 in F different from 0, such that for all a belonging to F, a * 1 = a.

Existence of additive inverses

For every a belonging to F, there exists an element -a in F, such that a + (-a) = 0.

Existence of multiplicative inverses

For every a ≠ 0 belonging to F, there exists an element a^-1 in F, such that a * a^-1 = 1.

The requirement 0 ≠ 1 ensures that the set which only contains a single zero is not a field. Directly from the axioms, one may show that (F, +) and (F - {0}, *) are commutative groupsIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G and that therefore (see elementary group theoryGroup theory First Theorems about Groups A group G ) is usually defined as: G is a set and is an associative binary operation on G obeying the following rules (or axioms): :A1. G ) has closure. That is, if a and b are in G then a ''b is in G :A2. The oper) the additive inverse -a and the multiplicative inverse a^-1 are uniquely determined by a. Furthermore, the multiplicative inverse of a product is equal to the product of the inverses:

(a*b)^-1 = b^-1 * a^-1 = a^-1 * b^-1

provided both a and b are non-zero. Other useful rules include

-a = (-1) * a

and more generally

-(a * b) = (-a) * b = a * (-b)

as well as

a * 0 = 0,

all rules familiar from elementary arithmeticArithmetic Arithmetic or arithmetics in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as synonym fo.

Politics	Business	Finance

Topics: Field (mathematics) Numbers Fields *the Set Theory Algebraic...

---

Field (mathematics)In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil Fields MedalThe Fields Medal is a prize awarded to up to four mathematicians (not over forty years of age) at each International Congress of International Mathematical Union, since 1936 and regularly since 1948 at the initiative of the Canadian mathematican John Char Field hockeyField hockey is a popular sport for men and women in many countries around the world. It is simply known as hockey in most countries, especially those in which ice hockey is not very prominent. Field hockey has several regular, prestigious international t Field-programmable gate arrayA field-programmable gate array or FPGA is a gate array that can be reprogrammed after it is manufactured, rather than having its programming fixed during the manufacturing — a programmable logic device. FPGAs are generally slower than their ASIC counterp Field effect transistorThe Field-Effect Transistor (FET) is a family of transistors that rely on an electric field to control the conductivity of a "channel" in a semiconductor material. FETs, like all transistors, can be thought of as voltage-controlled resistors. Most FETs ar Field extensionIn abstract algebra, an extension of a field K is a field L which contains K as a subfield. For example, C (the field of complex numbers) is an extension of R (the field of real numbers), and R is itself an extension of Q (the field of rational numbers). Field ion microscope Field strength Fielding (cricket) Field gun Field camera Field Emission Electric Propulsion Fieldon, Illinois Field Township, Minnesota Fieldon Township, Minnesota