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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.


DEFINITION

  • ) 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 ).


Spelled out, this means that the following hold:

  • : 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'').


; Existence of an additive identity : There exists an element 0 in ''F'', such that for all ''a'' belonging to ''F'', ''a'' + 0 = ''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.

  • ''a''−1 = 1.


  • ) 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''. Furthermore, the multiplicative inverse of a product is equal to the product of the inverses:

  • b'')−1 = ''b''−1 --- ''a''−1 = ''a''−1 --- ''b''−1

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

  • ''a''

    • and more generally

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

    • as well as

  • 0 = 0,

    • all rules familiar from elementary Arithmetic .


  • is dropped, one distinguishes the above commutative fields from '''non-commutative fields''', usually called Division Ring s or ''skew fields''.



EXAMPLES


  • The Complex Numbers \mathbb 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 "http://wwwinformationdelightinfo/encyclopedia/entry/characteristic_(algebra)" class="copylinks">Characteristic of any field is zero or a Prime Number (The characteristic is defined as follows: the smallest positive integer ''n'' such that ''n''·1 = 0, or zero if no such ''n'' exists here ''n''·1 stands for ''n'' summands 1 + 1 + 1 + + 1 An equivalent definition is the following: the characteristic of a field ''F'' is the unique non-negative generator of the kernel of the unique ring homomorphism '''Z''' &rarr ''F'' which sends 1 -> 1)