Mercury/God and Characteristic (algebra): Difference between pages

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In [[abstract algebra]], the '''characteristic''' of a [[mathematical ring|ring]] ''R'' is defined to be the smallest
#REDIRECT [[Mercury (God)]]
positive [[integer]] ''n'' such that 1<sub>''R''</sub>+...+1<sub>''R''</sub>
(with ''n'' summands) yields 0. If no such ''n'' exists, we say that the
characteristic of ''R'' is 0.

Alternatively, the characteristic of the ring ''R'' may be defined as that unique [[natural number]] ''n'' such that ''R'' contains a [[subring]] [[ring homomorphism|isomorphic]] to the factor ring '''Z'''/''n'''''Z'''.

'''Examples and notes:'''

* If ''R'' and ''S'' are rings and there exists a [[ring homomorphism]] ''R'' <tt>-></tt> ''S'', then the characteristic of ''S'' divides the characteristic of ''R''.
* For any [[integral domain]] (and in particular for any [[field]]), the characteristic is either 0 or [[prime number|prime]].
* For any [[ordered field]] (for example, the [[rational number|rationals]] or the [[real number|reals]]) the characteristic is 0.
* 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|complex numbers]] contain the rationals, their characteristic is 0.
* Any field of 0 characteristic is infinite. 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 functions over '''Z'''/''p'''''Z''' is one such. The [[algebraic closure]] of '''Z'''/''p'''''Z''' is another example.
* The size of any finite field of 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>. QED)
* If an integral domain ''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 map ''f''(''x'') = ''x''<sup>''p''</sup> defines a [[injective, surjective and bijective functions|injective]] [[ring homomorphism]] ''R'' <tt>-></tt> ''R''. It is called the ''Frobenius homomorphism''.
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<b>Characteristic</b> is also sometimes used as a piece of jargon in discussions of [[Universal--metaphysics|universals]] in [[metaphysics]], often in the phrase 'distinguishing characteristics'.

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[[talk:Characteristic|/Talk]]

Revision as of 08:55, 3 February 2002

In abstract algebra, the characteristic of a ring R is defined to be the smallest positive integer n such that 1R+...+1R (with n summands) yields 0. If no such n exists, we say that the characteristic of R is 0.

Alternatively, the characteristic of the ring R may be defined as that unique natural number n such that R contains a subring isomorphic to the factor ring Z/nZ.

Examples and notes:

  • If R and S are rings and there exists a ring homomorphism R -> S, then the characteristic of S divides the characteristic of R.
  • For any integral domain (and in particular for any field), the characteristic is either 0 or prime.
  • For any ordered field (for example, the rationals or the reals) the characteristic is 0.
  • 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.
  • Any field of 0 characteristic is infinite. 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 is one such. The algebraic closure of Z/pZ is another example.
  • The size of any finite field of 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. QED)
  • If an integral domain R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R. The map f(x) = xp defines a injective ring homomorphism R -> R. It is called the Frobenius homomorphism.

Characteristic is also sometimes used as a piece of jargon in discussions of universals in metaphysics, often in the phrase 'distinguishing characteristics'.


/Talk