Splitting Fields and Algebraic Closure

Adjoining one root of an irreducible polynomial gives a field where that polynomial acquires a single linear factor. Adjoining all of its roots makes it factor completely. The field this produces, its splitting field, is the central object Galois theory studies; iterating the idea over every polynomial at once yields the algebraic closure, a field in which nothing is left to factor.

Splitting fields

The minimality clause is what makes the splitting field a specific object rather than any field large enough to hold the roots. Concretely, a splitting field is generated over $F$ by the roots of $f$: if ${\alpha }_1,\dots ,{\alpha }_n$ are the roots in some extension where $f$ splits, then $F({\alpha }_1,\dots ,{\alpha }_n)$ is the splitting field. A degree-$n$ polynomial has at most $n$ roots, and exactly $n$ counted with multiplicity precisely when it splits completely.

Existence comes from iterating the single-root construction of the previous field lesson.

Adjoining a root drops the working degree by at least one and multiplies the running extension degree by at most that amount, which bounds the total.

The bound is often loose; the interesting examples are much smaller.

Worked splitting fields

Two easy cases bracket the range. The roots of $x^2-2$ are $\pm \sqrt{2}$, both in $\mathbb{Q}(\sqrt{2})$, so its splitting field is $\mathbb{Q}(\sqrt{2})$ of degree $2$; likewise $(x^2-2)(x^2-3)$ splits over $\mathbb{Q}(\sqrt{2},\sqrt{3})$, degree $4$. The next two are the instructive ones.

Cyclotomic fields as a splitting field

The splitting field of $x^n-1$ over $\mathbb{Q}$ deserves its own name. Its roots are the $n$th roots of unity, the complex numbers $e^{2\pi ik/n}$ for $k=0,\dots ,n-1$, which sit at $n$ equally spaced points on the unit circle.

Under multiplication the $n$th roots of unity form a finite subgroup of $K^{\times }$, hence a cyclic group of order $n$. A generator is a primitive $n$th root of unity, written ${\zeta }_n$; the other primitive roots are ${\zeta }_n^a$ with $\gcd (a,n)=1$, so there are $\phi (n)$ of them.

For a prime $p$, the factorization $x^p-1=(x-1)(x^{p-1}+\cdots +x+1)$ identifies the minimal polynomial of ${\zeta }_p$ as ${\Phi }_p(x)=x^{p-1}+\cdots +x+1$, irreducible by an Eisenstein shift, so $[\mathbb{Q}({\zeta }_p):\mathbb{Q}]=p-1$. The general degree $[\mathbb{Q}({\zeta }_n):\mathbb{Q}]=\phi (n)$ needs the cyclotomic polynomials ${\Phi }_n$, taken up in the cyclotomic extensions lesson.

Uniqueness of splitting fields

Any two splitting fields for the same polynomial are the same field up to isomorphism, so the splitting field is legitimate. Proving it for an isomorphism of base fields also supplies the lifting property the Galois correspondence uses.

Taking $\phi$ to be the identity on $F$ gives uniqueness.

A splitting field that is algebraic over $F$ and splits an entire collection of polynomials is called a normal extension; the term reappears when the Galois correspondence matches normal extensions with normal subgroups.

Algebraic closure

Pushing the splitting-field idea to every polynomial at once produces a field where nothing is left to factor.

In an algebraically closed field every polynomial splits, not just has one root: peel off a linear factor, and the cofactor again has a root. Taking the algebraic closure is idempotent — $\overline{F}$ is itself algebraically closed. If $f\in \overline{F}[x]$ has a root $\alpha$, then $\alpha$ is algebraic over $\overline{F}$ and hence, by transitivity of algebraic extensions, over $F$, so $\alpha \in \overline{F}$ already.⁴

Existence takes more care, because the roots of infinitely many polynomials are not a priori inside any one field. Artin's construction sidesteps the bookkeeping by giving each polynomial its own variable.

For subfields of the complex numbers this abstract construction is unnecessary. The fundamental theorem of algebra says $\mathbb{C}$ is algebraically closed — a statement Galois theory will prove later — so $\mathbb{C}$ contains an algebraic closure of each of its subfields. In particular the field $\overline{\mathbb{Q}}$ of algebraic numbers inside $\mathbb{C}$ is an algebraic closure of $\mathbb{Q}$. The practical upshot is that every computation with elements algebraic over $\mathbb{Q}$ can be read as happening inside one fixed field $\mathbb{C}$, and composites of algebraic extensions are unambiguous once all of them are viewed inside a common algebraic closure.