# Field

# Field

A field is the name given to a pair of numbers and a set of operations that together satisfy several specific laws. A familiar example is the set of rational numbers and the operations addition and multiplication. An example of a set of numbers that is not a field is the set of integers. It is an “integral domain.” It is not a field because it lacks multiplicative inverses. Without multiplicative inverses, division may be impossible.

The elements of a field obey the following laws:

1.Closure laws: a + b and ab are unique elements in the field.

2.Commutative laws: a + b = b + a and ab = ba.

3.Associative laws: a + (b + c) = (a + b) + c and a(bc) = (ab)c.

4.Identity laws: there exist elements 0 and 1 such that a + 0 = a and a x 1 = a.

5.Inverse laws: for every a there exists an element –a such that a + (–a) ≠0, and for every a ≠ 0 there exists an element a^{−1} such that a x a^{−1} = 1.

6.Distributive law: a(b + c) = ab + ac.

Rational numbers (numbers that can be expressed as the ratio a/b of an integer a and a natural number b) obey all these laws. They obey closure because the rules for adding and multiplying fractions, a/b + c/d = (ad + cb)/bd and (a/b)(c/d) = (ac)/(bd), convert these operations into adding and multiplying integers that are closed. They are commutative and associative because integers are commutative and associative. The ratio 0/1 is an additive identity, and the ratio 1/1 is a multiplicative identity. The ratios a/b and –a/b are additive inverses, and a/b and b/a (a, b ≠0) are multiplicative inverses. The rules for adding and multiplying fractions, together with the distributive law for integers, make the distributive law hold for rational numbers as well. Because the rational numbers obey all the laws, they form a field.

The rational numbers constitute the most widely used field, but there are others. The set of real numbers is a field. The set of complex numbers (numbers of the form a + bi, where a and b are real numbers, and i^{2} = –1) is also a field.

Although all the fields named above have an infinite number of elements in them, a set with only a finite number of elements can, under the right circumstances, be a field. For example, the set constitutes a field when addition and multiplication are defined by these tables:

With such a small number of elements, one can check that all the laws are obeyed by simply running down all the possibilities. For instance, the symmetry of the tables show that the commutative laws are obeyed. Verifying associativity and distributivity is a little tedious, but it can be done. The identity laws can be verified by looking at the tables. Where things become interesting is in finding inverses, since the addition table has no negative elements in it, and the multiplication table, no fractions. Two additive inverses have to add up to 0. According to the addition table 1 + 1 is 0; so 1, curiously, is its own additive inverse. The multiplication table is less remarkable. Zero never has a multiplicative inverse, and even in ordinary arithmetic, 1 is its own multiplicative inverse, as it is here.

This example is not as outlandish as one might think. If one replaces 0 with “even” and 1 with “odd,” the resulting tables are the familiar parity tables for catching mistakes in arithmetic.

One interesting situation arises where an algebraic number such as = √2 is used. (An algebraic number is one which is the root of a polynomial equation.) If one creates the set of numbers of the form a + b√2, where a and b are rational, this set constitutes a field. Every sum, product, difference, or quotient (except, of course, (a + b√2)/0) can be expressed as a number in that form. In fact, when one learns to rationalize the denominator in an expression such as 1/(1 – √2) that is what is going on. The set of such elements therefore form another field which is called an “algebraic extension” of the original field.

### KEY TERMS

**Field** —A set of numbers and operations exemplified by the rational numbers and the operations of addition, subtraction, multiplication, and division.

**Integral domain** —A set of numbers and operations exemplified by the integers and the operations addition, subtraction, and multiplication.

## Resources

### BOOKS

Singh, Jagjit, *Great Ideas of Modern Mathematics.* New York: Dover Publications, 1959.

### OTHER

*Wolfram MathWorld.* “Field” <http://mathworld.wolfram.com/Field.html> (accessed November 24, 2006).

J. Paul Moulton

# field

field / fēld/ •
n. 1. an area of open land, esp. one planted with crops or pasture, typically bounded by hedges or fences: *a wheat field* *a field of corn.* ∎ a piece of land used for a particular purpose, esp. an area marked out for a game or sport: *a football field.* ∎ Baseball defensive play or the defensive positions collectively: *he is fast in the field and on the bases.* ∎ a large area of land or water completely covered in a particular substance, esp. snow or ice. ∎ an area rich in a natural product, typically oil or gas: *an oil field.* ∎ an area on which a battle is fought: *a field of battle.* ∎ an area on a flag with a single background color: *fifty white stars on a blue field.* ∎ a place where a subject of scientific study or artistic representation can be observed in its natural location or context. 2. a particular branch of study or sphere of activity or interest: *we talked to professionals in various fields.* ∎ Comput. a part of a record, representing an item of data. 3. (usu. the field) all the participants in a contest or sport: *he destroyed the rest of the field with a devastating injection of speed.* 4. Physics the region in which a particular condition prevails, esp. one in which a force or influence is effective regardless of the presence or absence of a material medium. ∎ the force exerted or potentially exerted in such an area: *the variation in the strength of the field.*•
v. 1. [intr.] Baseball play as a fielder. ∎ [tr.] catch or stop (the ball): *he fielded the ball cleanly, but threw it down the right-field line.* 2. [tr.] send out (a team or individual) to play in a game: *a high school that traditionally fielded mediocre teams.* ∎ (of a political party) nominate (a candidate) to run in an election: *a radical political party that is beginning to field candidates in local elections.* ∎ deploy (an army): *the small gulf sheikhdoms fielded 11,500 troops with the Saudis.* 3. [tr.] deal with (a difficult question, telephone call, etc.): *she has fielded five calls from salespeople.*•
adj. carried out or working in the natural environment, rather than in a laboratory or office: *field observations.* ∎ (of an employee or work) away from the home office; remote: *a field representative.* ∎ (of military equipment) light and mobile for use on campaign: *field artillery.* ∎ denoting a game played outdoors on a marked field.PHRASES: play the field inf. indulge in a series of sexual relationships without committing oneself to anyone.DERIVATIVES: field·er n.

# Field

# Field

A *field* is the name given to a pair of numbers and a set of operations which together satisfy several specific laws. A familiar example of a field is the set of rational numbers and the operations addition and **multiplication** . An example of a set of numbers that is not a field is the set of **integers** . It is an "integral domain." It is not a field because it lacks multiplicative inverses. Without multiplicative inverses, **division** may be impossible.

The elements of a field obey the following laws:

- Closure laws: a + b and ab are unique elements in the field.
- Commutative laws: a + b = b + a and ab = ba.
- Associative laws: a + (b + c) = (a + b) + c and a(bc) = (ab)c.
- Identity laws: there exist elements 0 and 1 such that a + 0 = a and a × 1 = a.
- Inverse laws: for every a there exists an element - a such that a + (-a) = 0, and for every a ≠ 0 there exists an element a-1 such that a × a-1 = 1.
- Distributive law: a(b + c) = ab + ac.

Rational numbers (which are numbers that can be expressed as the **ratio** a/b of an integer a and a natural number b) obey all these laws. They obey closure because the rules for adding and multiplying fractions, a/b + c/d = (ad + cb)/bd and (a/b)(c/d) = (ac)/(bd), convert these operations into adding and multiplying integers which are closed. They are commutative and associative because integers are commutative and associative. The ratio 0/1 is an additive identity, and the ratio 1/1 is a multiplicative identity. The ratios a/b and -a/b are additive inverses, and a/b and b/a (a, b ≠ 0) are multiplicative inverses. The rules for adding and multiplying fractions, together with the distributive law for integers, make the distributive law hold for rational numbers as well. Because the rational numbers obey all the laws, they form a field.

The rational numbers constitute the most widely used field, but there are others. The set of **real numbers** is a field. The set of **complex numbers** (numbers of the form a + bi, where a and b are real numbers, and i2 = -1) is also a field.

Although all the fields named above have an infinite number of elements in them, a set with only a finite number of elements can, under the right circumstances, be a field. For example, the set constitutes a field when addition and multiplication are defined by these tables:

With such a small number of elements, one can check that all the laws are obeyed by simply running down all the possibilities. For instance, the **symmetry** of the tables show that the commutative laws are obeyed. Verifying associativity and distributivity is a little tedious, but it can be done. The identity laws can be verified by looking at the tables. Where things become interesting is in finding inverses, since the addition table has no **negative** elements in it, and the multiplication table, no fractions. Two additive inverses have to add up to 0. According to the addition table 1 + 1 is 0; so 1, curiously, is its own additive inverse. The multiplication table is less remarkable. **Zero** never has a multiplicative inverse, and even in ordinary **arithmetic** , 1 is its own multiplicative inverse, as it is here.

This example is not as outlandish as one might think. If one replaces 0 with "even" and 1 with "odd," the resulting tables are the familiar **parity** tables for catching mistakes in arithmetic.

One interesting situation arises where an algebraic number such as √ 2 is used. (An algebraic number is one which is the root of a polynomial equation.) If one creates the set of numbers of the form a + b √ 2 , where a and b are rational, this set constitutes a field. Every sum, product, difference, or quotient (except, of course, (a + b √ 2)/0) can be expressed as a number in that form. In fact, when one learns to rationalize the denominator in an expression such as 1/(1 - √ 2 ) that is what is going on. The set of such elements therefore form another field which is called an "algebraic extension" of the original field.

J. Paul Moulton

## Resources

### books

Birkhoff, Garrett, and Saunders MacLane. *A Survey of Modern Algebra.* New York: Macmillan Co., 1947.

McCoy, Neal H. *Rings and Ideals.* Washington, DC: The Mathematical Association of America, 1948.

Singh, Jagjit, *Great Ideas of Modern Mathematics.* New York: Dover Publications, 1959.

Stein, Sherman K. *Mathematics, the Man-Made Universe.* San Francisco: W. H. Freeman, 1969.

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Field**—A set of numbers and operations exemplified by the rational numbers and the operations of addition, subtraction, multiplication, and division.

**Integral domain**—A set of numbers and operations exemplified by the integers and the operations addition, subtraction, and multiplication.

# field

**field ( data field)** An item of data consisting of a number of characters, bytes, words, or codes that are treated together, e.g. to form a number, a name, or an address. A number of fields make a record and the fields may be fixed in length or variable. The term came into use with punched card systems and a field size was defined in terms of a number of columns.

**2.**Normally a way of designating a portion of a word that has a specific significance or function within that word, e.g. an address field in an instruction word or a character field within a data word.

**3.**In mathematics, a commutative ring containing more than one element and in which every nonzero element has an inverse with respect to the multiplication operation. Apart from their obvious relationship to arithmetic involving numbers of various kinds, fields play a very important role in discussion about the analysis of algorithms. Results in this area mention the number of operations of a particular kind, and these operations are usually related to addition and multiplication of elements of some field.

# Field

# Field

The term *field* designates a variety of different, closely related concepts in mathematics and physics that have been carried over into everyday language to designate a context or region of influence. In geometry a field is a function that is defined (i.e., has values) at every point of a manifold (smooth continuous surface). Similarly, in physics a field (e.g., an electric, magnetic, or gravitational field) is a function describing a physical quantity (e.g., electric, magnetic, or gravitational influence or forces) at all points of a region of space and time. Sometimes the region that is under the influence of an electric, magnetic, gravitational, or other source or agent is also referred to as a field. A similar and almost equivalent definition of a field in physics, especially in contemporary physics, is as a continuous dynamical system, or a dynamical system with infinite degrees of freedom. Fields are essential to the description of physical phenomena, particularly of the interaction between particles or other physical entities, and to the quantitative and qualitative modeling of forces, especially those that act at a distance without any medium.

*See also* Field Theories; Gravitation; Physics, Quantum

william r. stoeger

# field

**field** Field of the Cloth of Gold the scene of a meeting between Henry VIII of England and Francis I of France near Calais in 1520, for which both monarchs erected elaborate temporary palaces, including a sumptuous display of golden cloth. Little of importance was achieved, although the meeting symbolized Henry's determination to play a full part in European dynastic politics.

fields have eyes and woods have ears one may always be spied on by unseen watchers or listeners; an urban equivalent is *walls have ears* (see wall). The saying is recorded from the early 13th century.

see also a fair field and no favour, out of left field at left1.

# field

**field** open land, piece of land used for pasture or tillage OE.; ground on which a battle is fought XIII. OE. *feld*, corr. to OS. *feld* (Du. *veld*), (OH)G. *feld* :- WGmc. **felþu*; ult. rel. to OE. *folde* earth. ground, OS. *folda*, ON. *fold* and further to Gr. *platús*, Skr. *pṛthú-* broad.

# Field

# Field

competitors in a sporting event; the runners in a horse race; a stretch or expanse.

*Examples:* field of benefits, 1577; of clouds, 1860; of cricketers, 1850; of hounds [hunting], 1806; of horses [racing], 1771; of huntsmen, 1806; of ignorance, 1847; of miracles, 1712; of raillery; of runners [in races]; of stars, 1608; of woes, 1590.

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#### NEARBY TERMS

**field**