Characteristic (algebra)
The characteristic is a measure algebra of a ring or body. It indicates the least number of required steps, in which one must add the multiplicative identity element (1 ) of a body or ring to receive the additive identity element (0). If this is not possible, the characteristic is 0 to distinguish these, is the mathematical term character.
- 2.1 For rings 2.1.1 Examples
- 2.2.1 Examples
Definition
The characteristics of a unitary ring is the smallest natural number for which the arithmetic of the ring, the n- times the sum of the fuel element, the element is equal to zero, so
Each finite sum of ones equal to zero ( as for example in the case of real numbers ), then the ring is assigned to the characteristic definition.
A common abbreviation of the characteristics of being.
Alternative definition options are:
- The characteristic of the unitary ring is uniquely determined non-negative generator of the core of the canonical unitary ring homomorphism.
- The characteristic of the unitary ring is the uniquely determined non-negative integer that contains the unitary part of a ring which is isomorphic to the residue class ring. (Note that is. )
Remark
The above definitions apply to the particular characteristics of bodies, because everyone's body is a unitary ring.
Properties
For rings
Each unitary part of a unitary annular ring having the same characteristic as.
Is there a ring homomorphism between two unitary rings and so is the characteristic of a divisor of the characteristic of.
For each integral domain (and in particular every body ) is the characteristic either 0 or a prime number. In the latter case one speaks of positive characteristic.
Is a unitary ring with prime characteristic p, then for all. The picture is then a ring homomorphism and is called Frobeniushomomorphismus.
Examples
The residue class ring has the characteristic n
Since the field of complex numbers contains the rational numbers, so is his characteristic 0
For an irreducible polynomial g of degree n over the residual body, the factor is a ring body (which is isomorphic to the finite field ) which contains, and thus the characteristic is p.
When bodies
Every ordered field has the characteristic 0; Examples are the rational numbers or the real numbers. Each field of characteristic 0 is infinite; as it contains a prime field, which is isomorphic to the field of rational numbers.
Examples
There are infinite body with prime characteristic; Examples are the field of rational functions or the algebraic closure of.
The cardinality of a finite field of characteristic is a power of. Because in this case it contains the part of the body and is a finite dimensional vector space over this part of the body. From linear algebra it is known that the order of the vector space then is a power of.
It follows that each finite field as power comes at a prime power, since it then has to be a finite dimensional vector space over a finite field: Let the order of the finite part of the body and the dimension of the original body as a vector space over the part of the body. Then this vector space has many elements, which is a p- power.