1.14 Algebraically closed fields

Let $F$ be a field. A polynomial is said to split in $F[X]$ if it is a product of polynomials of degree at most $1$ in $F[X]$ .

Proposition 1.42

For a field $\Omega$ , the following statements are equivalent:

(a)

Every nonconstant polynomial in $\Omega[X]$ splits in $\Omega[X]$ .
(b)

Every nonconstant polynomial in $\Omega[X]$ has at least one root in $\Omega$ .
(c)

The irreducible polynomials in $\Omega[X]$ are those of degree $1$ .
(d)

Every field of finite degree over $\Omega$ equals $\Omega$ .

Proof.

The implications (a) $\Rightarrow$ (b) $\Rightarrow$ (c) are obvious.

(c) $\Rightarrow$ (d). Let $E$ be a finite extension of $\Omega$ , and let $\alpha\in E$ . The minimal polynomial of $\alpha$ , being irreducible, has degree $1$ , and so $\alpha\in\Omega$ .

(d) $\Rightarrow$ (c). Let $f$ be an irreducible polynomial in $\Omega[X]$ . Then $\Omega[X]/(f)$ is an extension of $\Omega$ of degree $\deg(f)$ (see 1.30), and so $\deg(f)=1$ . _□

Definition 1.43

(a) A field $\Omega$ is algebraically closed if it satisfies the equivalent statements of Proposition 1.42.

(b) A field $\Omega$ is an algebraic closure of a subfield $F$ if it is algebraically closed and algebraic over $F$ .

For example, the fundamental theorem of algebra (see LABEL:ag5 below) says that $\mathbb{C}{}$ is algebraically closed. It is an algebraic closure of $\mathbb{R}{}$ .

Proposition 1.44

If $\Omega$ is algebraic over $F$ and every polynomial $f\in F[X]$ splits in $\Omega[X]$ , then $\Omega$ is algebraically closed (hence an algebraic closure of $F$ ).

Proof.

Let $f$ be a nonconstant polynomial in $\Omega[X]$ . We have to show that $f$ has a root in $\Omega$ . We know (see 1.25) that $f$ has a root $\alpha$ in some finite extension $\Omega^{\prime}$ of $\Omega$ . Set

f=a_{n}X^{n}+\cdots+a_{0},\quad a_{i}\in\Omega,

and consider the fields

F\subset F[a_{0},\ldots,a_{n}]\subset F[a_{0},\ldots,a_{n},\alpha].

Each extension generated by a finite set of algebraic elements, and hence is finite (1.30). Therefore $\alpha$ lies in a finite extension of $F$ (see 1.20), and so is algebraic over $F$ — it is a root of a polynomial $g$ with coefficients in $F$ . By assumption, $g$ splits in $\Omega[X]$ , and so the roots of $g$ in $\Omega^{\prime}$ all lie in $\Omega$ . In particular, $\alpha\in\Omega.$ _□

Proposition 1.45

Let $\Omega\supset F$ ; then

\{\alpha\in\Omega\mid\alpha\text{{\ algebraic over }}F\}

is a field.

Proof.

If $\alpha$ and $\beta$ are algebraic over $F$ , then $F[\alpha,\beta]$ is a field (see 1.31) of finite degree over $F$ (see 1.30). Thus, every element of $F[\alpha,\beta]$ is algebraic over $F$ . In particular, $\alpha\pm\beta$ , $\alpha/\beta$ , and $\alpha\beta$ are algebraic over $F$ . _□

The field constructed in the proposition is called the algebraic closure of $F$ in $\Omega$ .

Corollary 1.46

Let $\Omega$ be an algebraically closed field. For any subfield $F$ of $\Omega$ , the algebraic closure $E$ of $F$ in $\Omega$ is an algebraic closure of $F.$

Proof.

It is algebraic over $F$ by definition. Every polynomial in $F[X]$ splits in $\Omega[X]$ and has its roots in $E$ , and so splits in $E[X]$ . Now apply Proposition 1.44. _□

Thus, when we admit the fundamental theorem of algebra (LABEL:ag5), every subfield of $\mathbb{C}{}$ has an algebraic closure (in fact, a canonical algebraic closure). Later (Chapter 6) we’ll prove, using the axiom of choice, that every field has an algebraic closure.

Aside 1.47

Although various classes of field, for example, number fields and function fields, had been studied earlier, the first systematic account of the theory of abstract fields was given by Steinitz in 1910 (Algebraische Theorie der Körper, J. Reine Angew. Math., 137:167–309). Here he introduced the notion of a prime field, distinguished between separable and inseparable extensions, and showed that every field can be obtained as an algebraic extension of a purely transcendental extension. He also proved that every field has an algebraic closure, unique up to isomorphism. His work influenced later algebraists (Emmy Noether, van der Waerden, Emil Artin, …) and his article has been described by Bourbaki as “… a fundamental work that may be considered as having given birth to the current conception¹³¹³ 13 In which objects are to be defined abstractly by axioms. of algebra”. See: Roquette, Peter, In memoriam Ernst Steinitz (1871–1928). J. Reine Angew. Math. 648 (2010), 1–11.