Characteristic (algebra)

Equivalent characterizations

• The characteristic of a ring ${\displaystyle R}$ is the natural number ${\displaystyle n}$ such that ${\displaystyle n\mathbb{Z}}$ is the kernel of the unique ring homomorphism from ${\displaystyle \mathbb{Z}}$ to ${\displaystyle R}$.
• The characteristic is the natural number ${\displaystyle n}$ such that ${\displaystyle R}$ contains a subring isomorphic to the factor ring ${\displaystyle \mathbb{Z}/n\mathbb{Z}}$, which is the image of the above homomorphism.
• When the non-negative integers ${\displaystyle \{0,1,2,3,\dots \}}$ are partially ordered by divisibility, then ${\displaystyle 1}$ is the smallest and ${\displaystyle 0}$ is the largest. Then the characteristic of a ring is the smallest value of ${\displaystyle n}$ for which ${\displaystyle n\cdot 1=0}$. If nothing "smaller" (in this ordering) than 0 will suffice, then the characteristic is ${\displaystyle 0}$. This is the appropriate partial ordering because of such facts as that ${\displaystyle \mathrm{char}(A\times B)}$ is the least common multiple of ${\displaystyle \mathrm{char}(A)}$ and ${\displaystyle \mathrm{char}(B)}$, and that no ring homomorphism ${\displaystyle f:A\to B}$ exists unless ${\displaystyle \mathrm{char}(B)}$ divides ${\displaystyle \mathrm{char}(A)}$.
• The characteristic of a ring ${\displaystyle R}$ is ${\displaystyle n}$ precisely if the statement ${\displaystyle ka=0}$ for all ${\displaystyle a\in R}$ holds if and only if ${\displaystyle k}$ is a multiple of ${\displaystyle n}$.

Case of rings

If R and S are rings and there exists a ring homomorphism R → S, then the characteristic of S divides the characteristic of R. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic 1 is the zero ring, which has only a single element 0. If a nontrivial ring R does not have any nontrivial zero divisors, then its characteristic is either 0 or prime. In particular, this applies to all fields, to all integral domains, and to all division rings. Any ring of characteristic zero is infinite.

The ring ${\displaystyle \mathbb{Z}/n\mathbb{Z}}$ of integers modulo n has characteristic n. If R is a subring of S, then R and S have the same characteristic. For example, if p is prime and q(X) is an irreducible polynomial with coefficients in the field ${\displaystyle {\mathbb{F}}_p}$ with p elements, then the quotient ring ${\displaystyle {\mathbb{F}}_p[X]/(q(X))}$ is a field of characteristic p. Another example: The field ${\displaystyle \mathbb{C}}$ of complex numbers contains ${\displaystyle \mathbb{Z}}$, so the characteristic of ${\displaystyle \mathbb{C}}$ is 0.

A ${\displaystyle \mathbb{Z}/n\mathbb{Z}}$-algebra is equivalently a ring whose characteristic divides n. This is because for every ring R there is a ring homomorphism ${\displaystyle \mathbb{Z}\to R}$, and this map factors through ${\displaystyle \mathbb{Z}/n\mathbb{Z}}$ if and only if the characteristic of R divides n. In this case for any r in the ring, then adding r to itself n times gives nr = 0.

If a commutative ring R has prime characteristic p, then we have (x + y)^p = x^p + y^p for all elements x and y in R – the normally incorrect "freshman's dream" holds for power p. The map x ↦ x^p then defines a ring homomorphism R → R, which is called the Frobenius homomorphism. If R is also an integral domain, the homomorphism is injective.

Case of fields

As mentioned above, the characteristic of any field is either 0 or a prime number. A field of non-zero characteristic is called a field of finite characteristic or positive characteristic or prime characteristic. The characteristic exponent is defined similarly, except that it is equal to 1 when the characteristic is 0; otherwise it has the same value as the characteristic.

Any field F has a unique minimal subfield, also called its prime field. This subfield is isomorphic to either the rational number field ${\displaystyle \mathbb{Q}}$ or a finite field ${\displaystyle {\mathbb{F}}_p}$ of prime order. Two prime fields of the same characteristic are isomorphic, and this isomorphism is unique. In other words, there is essentially a unique prime field in each characteristic.

Fields of characteristic zero

The fields of characteristic zero are those that have a subfield isomorphic to the field ⁠${\displaystyle \mathbb{Q}}$⁠ of the rational numbers. The most common of such fields are the subfields of the field ⁠${\displaystyle \mathbb{C}}$⁠ of the complex numbers; this includes the real numbers ${\displaystyle \mathbb{R}}$ and all algebraic number fields.

Other fields of characteristic zero are the p-adic fields that are widely used in number theory.

Fields of rational fractions over the integers or a field of characteristic zero are other common examples.

Ordered fields always have characteristic zero; they include ${\displaystyle \mathbb{Q}}$ and ${\displaystyle \mathbb{R}.}$

Fields of prime characteristic

The finite field ${\displaystyle {\mathbb{F}}_{p^n}}$ has characteristic ${\displaystyle p}$.

There exist infinite fields of prime characteristic. For example, the field of all rational functions over ${\displaystyle \mathbb{Z}/p\mathbb{Z}}$, the algebraic closure of ${\displaystyle \mathbb{Z}/p\mathbb{Z}}$ or the field of formal Laurent series ${\displaystyle \mathbb{Z}/p\mathbb{Z}((T))}$.

The size of any finite ring of prime characteristic ${\displaystyle p}$ is a power of ${\displaystyle p}$. Since in that case it contains ${\displaystyle \mathbb{Z}/p\mathbb{Z}}$ it is also 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.