Characteristic (algebra)

In algebra, the characteristic is an index of a ring or body . It indicates the smallest number of steps required, in which one has to add the multiplicative neutral element (1) of a body or ring to get the additive neutral element (0). If this is not possible, the characteristic is 0. A distinction must be made between this and the mathematical term character .

definition

The characteristic of a unitary ring is the smallest natural number for which in the arithmetic of the ring the -fold sum of the one element equals the zero element, i.e. ${\displaystyle R}$ ${\displaystyle n\geq 1}$ ${\displaystyle n}$ ${\displaystyle 1}$

${\displaystyle \underbrace {1+1+\dotsb +1} _{n{\text{-mal}}}=0}$,

if such a number exists. Otherwise, when every finite sum of ones is not equal to zero, the characteristic of the ring is defined as. ${\displaystyle 0}$

A common abbreviation for the characteristic of is . ${\displaystyle R}$ ${\displaystyle \operatorname {char} (R)}$

Alternative definition options that do not require any special treatment for the result are: ${\displaystyle 0}$

• The characteristic of the unitary ring is the uniquely determined nonnegative generator of the kernel of the canonical unitary ring homomorphism ${\displaystyle R}$

${\displaystyle \chi \colon \mathbb {Z} \to R\colon \chi (n):=n\cdot 1}$.

• The characteristic of the unitary ring is the uniquely determined nonnegative integer for which contains a unitary partial ring that is isomorphic to the remainder class ring . (Note that is.) ${\displaystyle R}$ ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ ${\displaystyle \mathbb {Z} /0\mathbb {Z} =\mathbb {Z} }$

comment

The above definitions also explain the characteristics of bodies, because every body is a unitary ring.

properties

With rings

Each unitary sub-ring of a unitary ring has the same characteristics as . ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle R}$

If there is a ring homomorphism between two unitary rings and , then the characteristic of is a factor of the characteristic of . ${\displaystyle R\to S}$ ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle R}$

For every integrity ring (and especially every body ) the characteristic is either 0 or a prime number . In the latter case, one speaks of a positive characteristic.

If a commutative unitary ring with prime number characteristics then applies to all . The mapping is then a ring homomorphism and is called a Frobenius homomorphism . ${\displaystyle R}$ ${\displaystyle p}$ ${\displaystyle (x+y)^{p}=x^{p}+y^{p}}$ ${\displaystyle x,y\in R}$ ${\displaystyle f\colon R\to R,\;x\mapsto x^{p}}$

A commutative ring with characteristic 0 is called a mixed characteristic ring if there is an ideal of the ring such that it has positive characteristics. An example is the ring of whole numbers with characteristic zero, in which for every prime number there is a finite field with characteristic . ${\displaystyle I}$ ${\displaystyle R/I}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle p}$ ${\displaystyle p}$

example

The residual class ring has the characteristic . ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ ${\displaystyle n}$

With bodies

Every ordered body has the characteristic 0; Examples are the rational numbers or the real numbers . Every field of characteristic 0 is infinite; namely, it contains a prime field that is isomorphic to the field of rational numbers.

Examples

• Since the field of complex numbers contains the rational numbers, its characteristic is also 0.

• For an irreducible polynomial of degree above the remainder class field , the factor ring is a field (which is isomorphic to the finite field ) which contains and therefore has the characteristic . ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle \mathbb {F} _{p}}$ ${\displaystyle \mathbb {F} _{p}[X]/(g)}$ ${\displaystyle \mathbb {F} _{p^{n}}}$ ${\displaystyle \mathbb {F} _{p}}$ ${\displaystyle p}$

• The thickness of a finite field of the characteristic is a power of . Because in it contains the part body and is a finite-dimensional vector space over this part body. It is known from linear algebra that the order of the vector space is then a power of . ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle \mathbb {F} _{p}}$ ${\displaystyle p}$

• There are infinite fields with prime number characteristics; Examples are the body of the rational functions over or the algebraic closure of . ${\displaystyle \mathbb {F} _{p}}$ ${\displaystyle \mathbb {F} _{p}}$

literature

• Siegfried Bosch : Algebra. 7th edition. Springer-Verlag, Berlin 2009, ISBN 978-3-540-92811-9 , section 3.1 ( limited preview in the Google book search).

Individual evidence

1. ↑ Mixed Characteristic , ncatlab