Inmathematics, thecharacteristic of aringR, often denotedchar(R), is defined to be the smallest positive number of copies of the ring'smultiplicative identity (1) that will sum to theadditive identity (0). If no such number exists, the ring is said to have characteristic zero.
That is,char(R) is the smallest positive numbern such that:[1](p 198, Thm. 23.14)
if such a numbern exists, and0 otherwise.
The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately.
The characteristic may also be taken to be theexponent of the ring'sadditive group, that is, the smallest positive integern such that:[1](p 198, Def. 23.12)
for every elementa of the ring (again, ifn exists; otherwise zero). This definition applies in the more general class ofrngs (seeRing (mathematics) § Multiplicative identity and the term "ring"); for (unital) rings the two definitions are equivalent due to theirdistributive law.
IfR andS arerings and there exists aring homomorphismR →S, then the characteristic ofS divides the characteristic ofR. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic1 is thezero ring, which has only a single element0. If a nontrivial ringR does not have any nontrivialzero divisors, then its characteristic is either0 orprime. In particular, this applies to allfields, to allintegral domains, and to alldivision rings. Any ring of characteristic zero is infinite.
The ring of integersmodulon has characteristicn. IfR is asubring ofS, thenR andS have the same characteristic. For example, ifp is prime andq(X) is anirreducible polynomial with coefficients in the field withp elements, then thequotient ring is a field of characteristicp. Another example: The field ofcomplex numbers contains, so the characteristic of is0.
A-algebra is equivalently a ring whose characteristic dividesn. This is because for every ringR there is a ring homomorphism, and this map factors through if and only if the characteristic ofR dividesn. In this case for anyr in the ring, then addingr to itselfn times givesnr = 0.
If a commutative ringR hasprime characteristicp, then we have(x +y)p =xp +yp for all elementsx andy inR – the normally incorrect "freshman's dream" holds for powerp.The mapx ↦xp then defines aring homomorphismR →R, which is called theFrobenius homomorphism. IfR is anintegral domain it isinjective.
As mentioned above, the characteristic of anyfield is either0 or a prime number. A field of non-zero characteristic is called a field offinite characteristic orpositive characteristic orprime characteristic. Thecharacteristic exponent is defined similarly, except that it is equal to1 when the characteristic is0; otherwise it has the same value as the characteristic.[2]
Any fieldF has a unique minimalsubfield, also called itsprime field. This subfield is isomorphic to either therational number field or a finite field of prime order. Two prime fields of the same characteristic are isomorphic, and this isomorphism is unique. In other words, there is essentially a unique prime field in each characteristic.
The fields ofcharacteristic zero are those that have a subfield isomorphic to the field of therational numbers. The most common ones are the subfields of the field of thecomplex numbers; this includes thereal numbers and allalgebraic number fields.
Other fields of charateristic zero are thep-adic fields that are widely used in number theory.
Fields ofrational fractions over the integers or a field of characteristic zero are other common examples.
Ordered fields have always the characteristic zero; they include and
Thefinite fieldGF(pn) has characteristicp.
There exist infinite fields of prime characteristic. For example, the field of allrational functions over, thealgebraic closure of or the field offormal Laurent series.
The size of anyfinite ring of prime characteristicp is a power ofp. Since in that case it contains it is also avector space over that field, and fromlinear 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.[b]
