Characteristic (algebra): Difference between revisions
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* 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''. |
* 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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See also: [[property--metaphysics]] |
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Revision as of 20:20, 17 January 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'.
Oy--we can look forward to moving this to characteristic (abstract algebra)