- #1

- 19,464

- 10,078

**Definition/Summary**A field is a commutative division ring.

That is, a commutative ring (a group under addition, and with multiplication, a multiplicative identity, and associative and distributive rules) in which division (the inverse of multiplication) is defined, except for division by zero.

So the non-zero elements of a field form a group under multiplication.

Note this is not the same notion as, say, an electromagnetic field in physics.

**Equations**A ring is a set, closed under operations [itex]+[/itex] and [itex]\cdot[/itex], with identity elements 0 and 1 respectively, with inverses under [itex]+[/itex], and with the following rules:

Associative:

[tex]a\ +\ (b\ +\ c)\ =\ (a\ +\ b)\ +\ c[/tex]

[tex]a\ \cdot\ (b\ \cdot\ c)\ =\ (a\ \cdot\ b)\ \cdot\ c[/tex]

Distributive:

[tex]a\ \cdot\ (b\ +\ c)\ =\ (a\ \cdot\ b)\ +\ (a\cdot\ c)[/tex]

[tex](b\ +\ c)\ \cdot\ a\ =\ (b\ \cdot\ a)\ +\ (c\cdot\ a)[/tex]

A commutative ring also has the rules:

Commutative:

[tex]a\ +\ b\ =\ b\ +\ a[/tex]

[tex]a\ \cdot\ b\ =\ b\ \cdot\ a[/tex]

**Extended explanation****Fields**

**Conventions and definitions**

It is traditional to denote a generic field by [itex]\mathbb{F}[/itex], or [itex]k[/itex]. We shall use [itex]k[/itex] in this topic.

**Definition**

A field [itex]k[/itex] is a commutative ring where the non-zero elements form a group under multiplication. By convention fields must have at least two distinct elements: [itex]0[/itex] the additive identity; [itex]1[/itex] the multiplicative identity.

**Examples**

Examples abound since fields are some of the most important objects in mathematics.

1) The rational numbers [itex]\mathbb{Q}[/itex]

2) The real numbers [itex]\mathbb{R}[/itex]

3) The complex numbers [itex]\mathbb{C}[/itex]

4) For each prime [itex]p[/itex] in [itex]\mathbb{N}[/itex], the finite field [itex]\mathbb{F}_p[/itex], or [itex]GF(p)[/itex].

5) The algebraic numbers: the set of all roots of polynomial equations with rational coefficients.

**Field Extensions**

Given some field [itex]k[/itex], one frequently wishes to extend the field. An example of when might wish to do this is to allow more polynomials to have roots. For example, consider the field [itex]\mathbb{F}_p[/itex]. By Fermat's little theorem, the polynomial

[tex]f(x)=x^p-x+1[/tex]

has no roots in the field [itex]\mathbb{F}_p[/itex]. We may declare [itex]\alpha[/itex] to be a symbol that satisfies [itex]f[/itex], and form the field [itex]\mathbb{F}_p[\alpha][/itex]. Elements of this extension are formal linear combinations (i.e. with coefficients in [itex]\mathbb{F}_p[/itex]) of powers of [itex]\alpha[/itex], with the rule that [itex]f(\alpha)=0[/itex].

Equivalently, one could define this as the quotient of a polynomial ring:

[tex]\frac{\mathbb{F}_p[x]}{x^p-x+1}[/tex].

If [itex]x^p-x+1[/itex] is irreducible (e.g. if [itex]p=2[/itex] then this is a field wth [itex]p^p[/itex] elements. One can construct fields with [itex]p^r[/itex] elements for any [itex]r[/itex] by quotienting the polyonomial ring by any irreducible poly of degree [itex]r[/itex]. Of course the reader must take on trust here that there is an irreducible polynomial of each degree. The field with [itex]p^r[/itex] elements is a subfield of the field with [itex]p^s[/itex] elements if and only if [itex]r[/itex] divides [itex]s[/itex].

**Number fields**

One important and well understood class of field extensions are

*number fields*. These merit a separate entry on their own.

**Algebraic closures**

**Definition**

A field [itex]k[/itex] is algebraically closed if any polyomial with coefficients in [itex]k[/itex] has a root in [itex]k[/itex].

*Remark*

Note, by the euclidean algorithm this implies that any polynomial has all its roots in [itex]k[/itex], and is the product (essentially uniquely) of linear factors with coefficients in [itex]k[/itex]. In particular, over an algebraically closed field a polynomial of degree [itex]r[/itex] has [itex]r[/itex] roots (with multiplicity). Contrast this to the ring [itex]\mathbb{Z}/8\mathbb{Z}[/itex] where the polynomial [itex]x^2-1[/itex] has more than 2 roots: 1,3,5 and 7 are all square roots of 1.

**Examples**

The complex numbers, and algebraic numbers are algebraically closed. No other field in our list of examples is algebraically closed.

Given an arbitrary field, [itex]k[/itex], we define its algebraic closure to be the smallest algebraically closed field containing [itex]k[/itex]. It is not clear

*a priori*that one can always form the algebraic closure of [itex]k[/itex]. Indeed, one needs to appeal to the axiom of choice in general.

Note that there are no algebraically closed finite fields.

**Characteristic of a field**

The examples given above of [itex]\mathbb{F}_p[/itex] are so-called fields of positive characteristic.

**Definition**

Let [itex]k[/itex] be a field. Its characteristic is the minimal integer [itex]m[/itex] such that [itex]1[/itex] added to itself [itex]m[/itex] times is equal to [itex]0[/itex]. If no such [itex]m[/itex] exists, we say [itex]k[/itex] has characteristic zero.

It is usually an introductory exercise to show that [itex]m[/itex] is prime if it is non-zero.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!