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{{wikibooks|Discrete Mathematics|Finite fields}} | |||
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. | |||
That is, char(''R'') is the smallest positive number ''n'' such that | |||
:<math>\underbrace{1+\cdots+1}_{n \text{ summands}} = 0</math> | |||
if such a number ''n'' exists, and 0 otherwise. | |||
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 | |||
= | :<math>\underbrace{a+\cdots+a}_{n \text{ summands}} = 0</math> | ||
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. | |||
== | == Other equivalent characterizations == | ||
* 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''; | |||
* 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. | |||
* 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''. | |||
* 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''. | |||
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). | |||
==Case of rings== | |||
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. | |||
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. | |||
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''. | |||
The map | |||
:''f''(''x'') = ''x''<sup>''p''</sup> | |||
then defines a [[ring homomorphism]] | |||
:''R'' → ''R''. | |||
It is called the ''[[Frobenius homomorphism]]''. If ''R'' is an [[integral domain]] it is [[injective]]. | |||
==Case of fields==<!-- This section is linked from [[Elliptic curve cryptography]] --> | |||
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''. | |||
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'' → ∞. | |||
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)). | |||
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>.) | |||
==See also== | |||
*[[Characteristic exponent of a field]] | |||
==References== | |||
* Neal H. McCoy (1964, 1973) ''The Theory of Rings'', [[Chelsea Publishing]], page 4. | |||
[[Category:Ring theory]] | |||
[[Category:Field theory]] | |||
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In mathematics, the characteristic of a 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.
That is, char(R) is the smallest positive number n such that
if such a number n exists, and 0 otherwise.
The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive n such that
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), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.
Other equivalent characterizations
- The characteristic is the natural number n such that nZ is the kernel of a ring homomorphism from Z to R;
- The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring Z/nZ, which would be the image of that homomorphism.
- 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.
- 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.
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).
Case of rings
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 divisors, then its characteristic is either 0 or prime. In particular, this applies to all fields, to all integral domains, and to all division rings. Any ring of characteristic 0 is infinite.
The ring Z/nZ of integers 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/pZ where p is prime, then the factor ring (Z/pZ)[X]/(q(X)) is a field of characteristic p. Since the complex numbers contain the rationals, their characteristic is 0.
If a commutative ring R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R – the "freshman's dream" holds for power p.
The map
- f(x) = xp
then defines a ring homomorphism
- R → R.
It is called the Frobenius homomorphism. If R is an integral domain it is injective.
Case of fields
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.
For any field F, there is a minimal subfield, namely the Template:Visible anchor, the smallest subfield containing 1F. It is isomorphic either to the rational number field Q, or a finite field of prime order, Fp; 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 numbers (unless they have very large cardinality, that is; in fact, any field of characteristic zero and cardinality at most continuum is isomorphic to a subfield of complex numbersPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. The p-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic pk, as k → ∞.
For any ordered field, as the field of rational numbers Q or the field of real numbers R, the characteristic is 0. Thus, 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(pn) has characteristic p. There exist infinite fields of prime characteristic. For example, the field of all rational functions over Z/pZ, the algebraic closure of Z/pZ or the field of formal Laurent series Z/pZ((T)).
The size of any finite ring of prime characteristic p is a power of p. Since in that case it must contain Z/pZ 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 pn. So its size is (pn)m = pnm.)
See also
References
- Neal H. McCoy (1964, 1973) The Theory of Rings, Chelsea Publishing, page 4.