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.

Quotient ring
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