Is There an Algebraic Closure for Every Field?

[Greg Bernhardt]

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 $+$ and $\cdot$, with identity elements 0 and 1 respectively, with inverses under $+$, and with the following rules:

Associative:
$$a\ +\ (b\ +\ c)\ =\ (a\ +\ b)\ +\ c$$
$$a\ \cdot\ (b\ \cdot\ c)\ =\ (a\ \cdot\ b)\ \cdot\ c$$

Distributive:
$$a\ \cdot\ (b\ +\ c)\ =\ (a\ \cdot\ b)\ +\ (a\cdot\ c)$$
$$(b\ +\ c)\ \cdot\ a\ =\ (b\ \cdot\ a)\ +\ (c\cdot\ a)$$

A commutative ring also has the rules:

Commutative:
$$a\ +\ b\ =\ b\ +\ a$$
$$a\ \cdot\ b\ =\ b\ \cdot\ a$$

Extended explanation

Fields

Conventions and definitions

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

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

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

1) The rational numbers $\mathbb{Q}$

2) The real numbers $\mathbb{R}$

3) The complex numbers $\mathbb{C}$

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

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

Field Extensions

Given some field $k$, 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 $\mathbb{F}_p$. By Fermat's little theorem, the polynomial

$$f(x)=x^p-x+1$$

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

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

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

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

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 $k$ is algebraically closed if any polyomial with coefficients in $k$ has a root in $k$.

Remark
Note, by the euclidean algorithm this implies that any polynomial has all its roots in $k$, and is the product (essentially uniquely) of linear factors with coefficients in $k$. In particular, over an algebraically closed field a polynomial of degree $r$ has $r$ roots (with multiplicity). Contrast this to the ring $\mathbb{Z}/8\mathbb{Z}$ where the polynomial $x^2-1$ 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, $k$, we define its algebraic closure to be the smallest algebraically closed field containing $k$. It is not clear a priori that one can always form the algebraic closure of $k$. 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 $\mathbb{F}_p$ are so-called fields of positive characteristic.

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

It is usually an introductory exercise to show that $m$ is prime if it is non-zero.

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[fresh_42]

Interesting fields are also the p-adic numbers here. They have characteristic zero and their algebraic closure is of infinite degree, other than $[\mathbb{C}:\mathbb{R}]=2$.

We can always find an algebraic closure if we adjoint all roots of polynomials, as we did to define the complex numbers. But $\mathbb{C} \cong \mathbb{R}[x]/ \langle x^2+1\rangle$ is certainly the most important one.