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 same rules hold which are familiar from the Arithmetic of ordinary Number s.

All fields are Rings , but not conversely. Fields differ from rings most importantly in the requirement that division be possible, but also, in modern definitions, by the requirement that the multiplication operation in a field be Commutative . Otherwise the structure is a so-called ''skew field'' (better known as a Division Ring ), although historically division rings were called ''fields'' and fields were ''commutative fields''.

The prototypical example of a field is Q, the field of Rational Number s. Other important examples include the field of Real Number s '''R''', the field of Complex Number s '''C''' and, for any Prime Number ''p'', the Finite Field of Integers Modulo ''p'' , denoted '''Z'''/''p'''''Z''', F_''p'' or GF(''p''). For any field ''K'', the set ''K''(''X'') of Rational Functions with coefficients in ''K'' is also a field.

The mathematical discipline concerned with the study of fields is called Field Theory .

EQUIVALENT DEFINITIONS

Definition 1

A ''field'' is a commutative Division Ring .

Definition 2

) such that 0 does not equal 1 and all elements of ''F'' except 0 have a multiplicative inverse. (Note that 0 and 1 here stand for the identity elements for the + and --- operations respectively, which may differ from the familiar real numbers 0 and 1 ).

Definition 3

Explicitly, a field is defined by these properties:

: 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'').

are associative : For all ''a'', ''b'', ''c'' in ''F'', ''a'' + (''b'' + ''c'') = (''a'' + ''b'') + ''c'' and ''a'' --- (''b'' --- ''c'') = (''a'' --- ''b'') --- ''c''.

are commutative : For all ''a'', ''b'' belonging to ''F'', ''a'' + ''b'' = ''b'' + ''a'' and ''a'' --- ''b'' = ''b'' --- ''a''.

is distributive over the operation + : For all ''a'', ''b'', ''c'', belonging to ''F'', ''a'' --- (''b'' + ''c'') = (''a'' --- ''b'') + (''a'' --- ''c'').

1 = ''a''.

''a''⁻¹ = 1.

) are commutative Groups ( Abelian Group s) 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

''a''

b'') = (−''a'') --- b = ''a'' --- (−''b'')

0 = 0,

Arithmetic

is dropped, one distinguishes the above commutative fields from '''non-commutative fields'''. Fields which are not assumed to be commutative are usually called Division Ring s or ''skew fields''.

HISTORY

The concept of a field is due to Dedekind , who used the
word ''Körper'' "body" for this notion. He also was the first
to define rings (then called ''order'' or ''order-modul''),
but the term ''"a ring"'' (''Zahlring'') was invented by Hilbert . J J O'Connor and E F Robertson, ''The development of Ring Theory'' , September 2004.

EXAMPLES

The Complex Number s C, under the usual operations of addition and multiplication. The field of complex numbers contains the following ''subfields'' (a subfield of a field ''F'' is a set containing 0 and 1, closed under the operations + , - and --- of ''F'' and with its own operations defined by restriction):

The Real Number s R, under the usual operations of addition and multiplication. When the real numbers are given the usual ordering, they form a ''complete Ordered Field '' ; it is this structure which provides the foundation for most formal treatments of Calculus .

---The real numbers contain several interesting subfields: the real Algebraic Number s, the Computable Number s.

There is ( Up To Isomorphism ) exactly one Finite Field with ''q'' elements, for every finite number ''q'' which is a power of a Prime Number , ''q''≠ 1. (No finite field can exist with any other number of elements.) This is usually denoted F_''q'' . Such fields are often called Galois Field s.

---In particular, for a given prime number ''p'', the set of integers modulo ''p'' is a finite field with ''p'' elements: Z/''p''Z = '''F'''_''p'' = {0, 1, ..., ''p'' − 1} where the operations are defined by performing the operation in Z, dividing by ''p'' and taking the remainder; see Modular Arithmetic .

---Taking ''p'' = 2, we obtain the smallest field, F₂, which has only two elements: 0 and 1. It can be defined by the two Cayley Table s

0 '''1'''

1 1 0 1 0 1

::This field has important uses in Computer Science , especially in Cryptography and Coding Theory .

The rational numbers can be extended to the fields of ''p''-adic Numbers for every prime number ''p''. These fields are very important in both Number Theory and Mathematical Analysis .

Let ''E'' and ''F'' be two fields with ''F'' a subfield of ''E''. Let ''x'' be an element of ''E'' not in ''F''. Then there is a smallest subfield of ''E'' containing ''F'' and ''x'', denoted ''F''(''x''). We call ''F''(''x'') a '' Simple Extension '' of ''F''. For instance, Q(''i'') is the subfield of '''C''' consisting of all numbers of the form ''a'' + ''bi'' where both ''a'' and ''b'' are rational numbers. In fact, it can be shown that every number field is a simple extension of Q.

For a given field ''F'', the set ''F''(''X'') of Rational Function s in the variable ''X'' with coefficients in ''F'' is a field; this is the Quotient Field of the ring of Polynomials ''F'' {Link without Title} . This is the simplest example of a Transcendental Extension of ''F''.

If ''F'' is a field, and ''p''(''X'') is an Irreducible Polynomial in the Polynomial Ring ''F'' then the quotient ''F''[''X'' /<''p''(''X'')> , where <''p''(''X'')> denotes the ideal generated by ''p''(''X''), is a field with a subfield isomorphic to ''F''. For instance, R[''X'']/<''X''² + 1> is a field (in fact, it is isomorphic to the field of complex numbers). It can be shown that every simple algebraic extension of ''F'' is isomorphic to a field of this form. See the Primitive Element Theorem .

When ''F'' is a field, the set ''F''((''X'')) of Formal Laurent Series over ''F'' is a field.

If ''V'' is an Algebraic Variety over ''F'', then the rational functions ''V'' → ''F'' form a field, the ''function field'' of ''V''.

If ''S'' is a Riemann Surface , then the Meromorphic Function s ''S'' → C form a field.

If ''I'' is an index set, ''U'' is an Ultrafilter on ''I'', and ''F''_''i'' is a field for every ''i'' in ''I'', the Ultraproduct of the ''F''_''i'' with respect to ''U'' is a field.

Hyperreal Numbers and Superreal Number s extend the real numbers with the addition of infinitesimal and infinite numbers.

There are also proper classes with field structure, which are sometimes called Fields, with a capital F:

The Surreal Number s form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. The set of all surreal numbers with birthday smaller than some Inaccessible Cardinal form a field.

The Nimber s form a Field. The set of nimbers with birthday smaller than $2^{2^n}$ , the nimbers with birthday smaller than any infinite Cardinal are all examples of fields.

SOME FIRST THEOREMS

The set of non-zero elements of a field ''F'' (typically denoted by ''F''^×) is an Abelian Group under multiplication. Every finite subgroup of ''F''^× is Cyclic .

The number of elements of any Finite Field is a prime power.

If there are positive integers ''n'' such that 0 = 1 + 1 + ... + 1 (''n'' repeated terms), then the smallest such ''n'' must be a prime number; that is, the Characteristic of a field must be either a prime number, or zero.

A commutative Ring is a field if and only if it has no Ideal s except {0} and itself.

Assuming the Axiom Of Choice , for every field ''F'', there exists a unique field ''G'', up to isomorphism inducing the identity on ''F'', which contains ''F'', is Algebraic over ''F'', and is Algebraically Closed . ''G'' is called the Algebraic Closure of ''F''. However, in many circumstances in mathematics, it is not appropriate to treat ''G'' as being uniquely determined by ''F'', since the isomorphism above is not itself unique. In these cases, one refers to such a ''G'' as ''an'' algebraic closure of ''F''.

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Field (mathematics)