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| {{wikibooks|Discrete Mathematics|Finite fields}}
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| In [[mathematics]], the '''characteristic''' of a [[ring (mathematics)|ring]] ''R'', often denoted char(''R''), is defined to be the smallest number of times one must use the ring's multiplicative [[identity element]] (1) in a sum to get the [[additive identity]] element (0); the ring is said to have characteristic zero if this sum never reaches the additive identity.
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| That is, char(''R'') is the smallest positive number ''n'' such that
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| :<math>\underbrace{1+\cdots+1}_{n \text{ summands}} = 0</math>
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| if such a number ''n'' exists, and 0 otherwise.
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| The characteristic may also be taken to be the [[exponent (group theory)|exponent]] of the ring's additive group, that is, the smallest positive ''n'' such that | |
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| :<math>\underbrace{a+\cdots+a}_{n \text{ summands}} = 0</math>
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| for every element ''a'' of the ring (again, if ''n'' exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (''see [[ring (algebra)|ring]]''), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the [[distributive law]] in rings.
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| == Other equivalent characterizations ==
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| * The characteristic is the [[natural number]] ''n'' such that ''n'''''Z''' is the [[kernel (ring theory)|kernel]] of a [[ring homomorphism]] from '''Z''' to ''R'';
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| * The characteristic is the [[natural number]] ''n'' such that ''R'' contains a [[subring]] [[ring homomorphism|isomorphic]] to the [[factor ring]] '''Z'''/''n'''''Z''', which would be the [[image (mathematics)|image]] of that homomorphism.
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| * When the non-negative integers {0, 1, 2, 3, . . . } are partially ordered by divisibility, then 1 is the smallest and 0 is the largest. Then the characteristic of a ring is the smallest value of ''n'' for which ''n'' · 1 = 0. If nothing "smaller" (in this ordering) than 0 will suffice, then the characteristic is 0. This is the right partial ordering because of such facts as that char ''A'' × ''B'' is the [[least common multiple]] of char ''A'' and char ''B'', and that no ring homomorphism ''ƒ'' : ''A'' → ''B'' exists unless char ''B'' divides char ''A''.
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| * The characteristic of a ring ''R'' is ''n'' ∈ {0, 1, 2, 3, . . . } precisely if the statement ''ka'' = 0 for all ''a'' ∈ ''R'' implies ''n'' is a divisor of ''k''.
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| The requirements of ring homomorphisms are such that there can be only one homomorphism from the ring of integers to any ring; in the language of [[category theory]], '''Z''' is an [[initial object]] of the [[category of rings]]. Again this follows the convention that a ring has a multiplicative identity element (which is preserved by ring homomorphisms).
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| ==Case of rings==
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| If ''R'' and ''S'' are rings and there exists a [[ring homomorphism]] ''R'' → ''S'', then the characteristic of ''S'' divides the characteristic of ''R''. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic 1 is the [[trivial ring]] which has only a single element 0 = 1. If a non-trivial ring ''R'' does not have any [[zero divisor]]s, then its characteristic is either 0 or [[prime number|prime]]. In particular, this applies to all [[field (mathematics)|fields]], to all [[integral domain]]s, and to all [[division ring]]s. Any ring of characteristic 0 is infinite.
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| The ring '''Z'''/''n'''''Z''' of integers [[modular arithmetic|modulo]] ''n'' has characteristic ''n''. If ''R'' is a [[subring]] of ''S'', then ''R'' and ''S'' have the same characteristic. For instance, if ''q''(''X'') is a prime [[polynomial]] with coefficients in the field '''Z'''/''p'''''Z''' where ''p'' is prime, then the factor ring ('''Z'''/''p'''''Z''')[''X'']/(''q''(''X'')) is a field of characteristic ''p''. Since the [[complex number]]s contain the rationals, their characteristic is 0.
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| If a commutative ring ''R'' has '''prime characteristic''' ''p'', then we have (''x'' + ''y'')<sup>''p''</sup> = ''x''<sup>''p''</sup> + ''y''<sup>''p''</sup> for all elements ''x'' and ''y'' in ''R'' – the "[[freshman's dream]]" holds for power ''p''.
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| The map
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| :''f''(''x'') = ''x''<sup>''p''</sup>
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| then defines a [[ring homomorphism]]
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| :''R'' → ''R''.
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| It is called the ''[[Frobenius homomorphism]]''. If ''R'' is an [[integral domain]] it is [[injective]].
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| ==Case of fields==<!-- This section is linked from [[Elliptic curve cryptography]] -->
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| As mentioned above, the characteristic of any field is either 0 or a prime number. A field of non-zero characteristic is called a field of ''finite characteristic'' or a field of ''positive characteristic''.
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| For any field ''F'', there is a minimal [[subfield]], namely the '''{{visible anchor|prime field}}''', the smallest subfield containing 1<sub>''F''</sup>. It is isomorphic either to the [[rational number]] field '''Q''', or a finite field of prime order, '''F'''<sub>''p''</sub>; the structure of the prime field and the characteristic each determine the other. Fields of ''characteristic zero'' have the most familiar properties; for practical purposes they resemble subfields of the [[complex number]]s (unless they have very large [[cardinality]], that is; in fact, any field of characteristic zero and cardinality at most [[Cardinality of the continuum|continuum]] is isomorphic to a subfield of complex numbers{{citation needed|date=September 2013}}. The [[p-adic field]]s or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic ''p''<sup>''k''</sup>, as ''k'' → ∞.
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| For any [[ordered field]], as the field of [[rational number]]s '''Q''' or the field of [[real number]]s '''R''', the characteristic is 0. Thus, [[algebraic number field|number fields]] and the field of complex numbers '''C''' are of characteristic zero. Actually, every field of characteristic zero is the quotient field of a ring '''Q'''[X]/P where X is a set of variables and P a set of polynomials in '''Q'''[X]. The [[finite field]] GF(''p''<sup>''n''</sup>) has characteristic ''p''. There exist infinite fields of prime characteristic. For example, the field of all [[rational function]]s over '''Z'''/''p'''''Z''', the [[algebraic closure]] of '''Z'''/''p'''''Z''' or the field of [[formal power series|formal Laurent series]] '''Z'''/''p'''''Z'''((T)).
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| The size of any [[finite ring]] of prime characteristic ''p'' is a power of ''p''. Since in that case it must contain '''Z'''/''p'''''Z''' it must also be a [[vector space]] over that field and from [[linear algebra]] we know that the sizes of finite vector spaces over finite fields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power. (It is a vector space over a finite field, which we have shown to be of size ''p''<sup>''n''</sup>. So its size is (''p''<sup>''n''</sup>)<sup>''m''</sup> = ''p''<sup>''nm''</sup>.)
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| ==See also==
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| *[[Characteristic exponent of a field]]
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| ==References==
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| * Neal H. McCoy (1964, 1973) ''The Theory of Rings'', [[Chelsea Publishing]], page 4.
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| [[Category:Ring theory]]
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| [[Category:Field theory]]
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