---
title: Splitting Fields and Algebraic Closure
module: Field Theory
moduleNumber: 10
lessonNumber: 3
order: 1003
summary: >
  The splitting field of a polynomial is the smallest extension in which it
  factors into linear pieces, obtained by adjoining all its roots. Every
  polynomial has one, its degree is at most n factorial, and any two splitting
  fields are isomorphic. Pushing this to all polynomials at once gives the
  algebraic closure, a field in which every polynomial splits and which is unique
  up to isomorphism.
topics: [Field Theory]
sources:
  - book: Dummit & Foote
    ref: "Ch. 13 — Field Theory; §13.4 Splitting Fields and Algebraic Closures"
draft: false
---

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

> **Definition (Splitting field).** An extension $K/F$ is a **splitting field**
> for $f(x) \in F[x]$ if $f$ factors into linear factors in $K[x]$ — it **splits
> completely** — and $f$ does not split completely over any proper subfield of
> $K$ containing $F$.

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](/abstract-algebra/field-theory/field-extensions-and-algebraic-elements).

> **Theorem (Existence).** For any field $F$ and any $f(x) \in F[x]$, there
> exists a splitting field for $f$ over $F$.

> **Proof.** Induct on $\deg f$. If every irreducible factor of $f$ is linear,
> $F$ already splits it. Otherwise an irreducible factor $p(x)$ has degree at
> least $2$; adjoin a root $\alpha$ to get $F_1 = F(\alpha)$, over which $f$ has
> the factor $x - \alpha$ and a cofactor of degree $\deg f - 1$. By induction the
> cofactor has a splitting field $E$ over $F_1$; intersecting the subfields of
> $E$ that contain $F$ and all roots of $f$ gives a splitting field.[^df-thm25]

$$
% caption: A splitting field is reached by adjoining roots one at a time, each
% step peeling off a linear factor until the polynomial factors completely.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\node[align=center] (s0) at (0,0) {F\\f irreducible\\factor of degree 3};
\node[align=center] (s1) at (4,0) {F(r1)\\one linear factor\\times a quadratic};
\node[align=center] (s2) at (8,0) {F(r1, r2)\\two linear factors\\times a line};
\node[draw=acc, thick, fill=acc!8, align=center] (s3) at (11.6,0) {K splits f\\into linear factors};
\draw[->, acc, thick] (s0) -- (s1) node[midway, above, font=\scriptsize] {adjoin r1};
\draw[->, acc, thick] (s1) -- (s2) node[midway, above, font=\scriptsize] {adjoin r2};
\draw[->, acc, thick] (s2) -- (s3) node[midway, above, font=\scriptsize] {adjoin r3};
\end{tikzpicture}
$$

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.

> **Proposition (Degree bound).** A splitting field of a degree-$n$ polynomial
> over $F$ has degree at most $n!$ over $F$.

> **Proof.** The first generator has degree at most $n$, the next at most
> $n - 1$ over the field so far, and so on, so the tower law gives a product
> bounded by $n(n-1)\cdots 1 = n!$.[^df-prop26]

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.

> **Worked example ($x^3 - 2$ over $\mathbb{Q}$).** The three roots are
> $\sqrt[3]{2}$, $\omega\sqrt[3]{2}$, $\omega^2\sqrt[3]{2}$, where
> $\omega = e^{2\pi i/3} = \tfrac{-1 + \sqrt{-3}}{2}$ is a primitive cube root of
> unity satisfying $x^2 + x + 1 = 0$. The real root alone gives
> $\mathbb{Q}(\sqrt[3]{2})$, a field of real numbers with
> $[\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3$, which cannot contain the complex
> roots. Adjoining any second root supplies the ratio $\omega$, so
> $$
> K = \mathbb{Q}(\sqrt[3]{2},\, \omega) = \mathbb{Q}(\sqrt[3]{2},\, \sqrt{-3}).
> $$
> Since $\sqrt{-3} \notin \mathbb{Q}(\sqrt[3]{2}) \subset \mathbb{R}$, its minimal
> polynomial $x^2 + 3$ stays irreducible there, so
> $[K : \mathbb{Q}(\sqrt[3]{2})] = 2$ and the tower law gives
> $$
> [K : \mathbb{Q}] = 2 \cdot 3 = 6,
> $$
> with $\mathbb{Q}$-basis $\{1, \sqrt[3]{2}, \sqrt[3]{4}\}$ times
> $\{1, \sqrt{-3}\}$. The bound $n! = 6$ is attained.

> **Worked example ($x^4 + 4$ over $\mathbb{Q}$).** The Sophie Germain identity
> factors it over $\mathbb{Q}$ before any extension:
> $$
> x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2).
> $$
> The quadratic formula gives the roots of the two factors as $-1 \pm i$ and
> $1 \pm i$, all four lying in $\mathbb{Q}(i)$. So the splitting field is
> $\mathbb{Q}(i)$ of degree $2$ — far below the bound $4! = 24$, because the
> polynomial was already reducible over the base.

$$
% caption: Known subfields of the degree-6 splitting field of x cubed minus 2:
% a degree-3 real branch and a degree-2 branch meet at the top.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\node (Q) at (0,0) {Q};
\node (R3) at (-2.2,1.6) {Q(w)};
\node (C3) at (2.2,1.6) {Q(c)};
\node[draw=acc, thick, fill=acc!8] (K) at (0,3.4) {K};
\draw[thick] (Q) -- (R3) node[midway, left=2pt, font=\footnotesize] {2};
\draw[thick] (Q) -- (C3) node[midway, right=2pt, font=\footnotesize] {3};
\draw[thick] (R3) -- (K) node[midway, left=2pt, font=\footnotesize] {3};
\draw[thick] (C3) -- (K) node[midway, right=2pt, font=\footnotesize] {2};
\node[font=\footnotesize, align=left, text=black] at (5.0,1.7)
  {c is the cube root of 2\\w is the square root of minus 3\\K has degree 6 over Q};
\end{tikzpicture}
$$

## 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
$e^{2\pi i k/n}$ for $k = 0, \dots, n-1$, which sit at $n$ equally spaced points
on the unit circle.

$$
% caption: The eighth roots of unity are eight equally spaced points on the unit
% circle; a primitive one generates the rest as its powers.
\begin{tikzpicture}[scale=1.2, >=stealth, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black] (0,0) circle (1.5);
\draw[black, ->] (-1.75,0) -- (1.75,0);
\draw[black, ->] (0,-1.75) -- (0,1.75);
\foreach \k in {0,1,2,3,4,5,6,7} {
  \fill[acc] (\k*45:1.5) circle (1.6pt);
}
\node[anchor=west, font=\footnotesize] at (1.55,-0.18) {1};
\node[acc, anchor=south west, font=\footnotesize] at (45:1.5) {generator};
\fill[acc] (45:1.5) circle (2.4pt);
\end{tikzpicture}
$$

Under multiplication the $n$th roots of unity form a finite subgroup of $K^\times$,
hence a [cyclic group](/abstract-algebra/subgroups-and-quotients/cyclic-groups)
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 $\varphi(n)$ of them.

> **Definition (Cyclotomic field).** The splitting field
> $\mathbb{Q}(\zeta_n)$ of $x^n - 1$ over $\mathbb{Q}$ is the **cyclotomic
> field** of $n$th roots of unity.

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
$[\mathbb{Q}(\zeta_n) : \mathbb{Q}] = \varphi(n)$ needs the cyclotomic
polynomials $\Phi_n$, taken up in the
[cyclotomic extensions](/abstract-algebra/field-theory/separable-and-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.

> **Theorem (Isomorphism extension).** Let $\varphi : F \to F'$ be a field
> isomorphism, let $f \in F[x]$, and let $f' \in F'[x]$ be its image under
> $\varphi$. If $E$ splits $f$ over $F$ and $E'$ splits $f'$ over $F'$, then
> $\varphi$ extends to an isomorphism $\sigma : E \to E'$.

> **Proof.** Induct on $\deg f$. If $f$ already splits over $F$, then $E = F$,
> $E' = F'$, and $\sigma = \varphi$. Otherwise pick an irreducible factor $p$ of
> $f$ with corresponding factor $p'$ of $f'$, a root $\alpha \in E$ of $p$, and a
> root $\beta \in E'$ of $p'$. The single-root extension theorem lifts $\varphi$
> to $\sigma' : F(\alpha) \to F'(\beta)$ sending $\alpha \mapsto \beta$. Over
> $F(\alpha)$ the polynomial $f$ loses the factor $x - \alpha$, and $E$ is a
> splitting field of the lower-degree cofactor, so induction extends $\sigma'$ to
> $\sigma : E \to E'$.[^df-thm27]

$$
% caption: The isomorphism-extension theorem lifts an isomorphism of base fields
% up to their splitting fields, one adjoined root at a time.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\node (E) at (0,3) {E};
\node (Ep) at (4.5,3) {E'};
\node (Fa) at (0,1.5) {F(r)};
\node (Fpb) at (4.5,1.5) {F'(s)};
\node (F) at (0,0) {F};
\node (Fp) at (4.5,0) {F'};
\draw[thick] (F) -- (Fa) -- (E);
\draw[thick] (Fp) -- (Fpb) -- (Ep);
\draw[->, black, thick] (F) -- (Fp) node[midway, above, font=\scriptsize] {phi};
\draw[->, black, thick] (Fa) -- (Fpb) node[midway, above, font=\scriptsize] {r maps to s};
\draw[->, acc, thick] (E) -- (Ep) node[midway, above, font=\scriptsize] {sigma};
\end{tikzpicture}
$$

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

> **Corollary (Uniqueness).** Any two splitting fields of $f(x) \in F[x]$ over
> $F$ are isomorphic by an isomorphism fixing $F$.

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.

> **Definition (Algebraically closed, algebraic closure).** A field $K$ is
> **algebraically closed** if every nonconstant polynomial in $K[x]$ has a root
> in $K$. An **algebraic closure** $\overline{F}$ of $F$ is an algebraic
> extension of $F$ in which every polynomial in $F[x]$ splits completely.

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.[^df-prop29]

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.

> **Proposition (Existence).** Every field $F$ has an algebraic closure, and it
> is unique up to isomorphism.

> **Proof.** For every nonconstant monic $f \in F[x]$ introduce an indeterminate
> $x_f$, and in the huge ring $F[\dots, x_f, \dots]$ form the ideal $I$ generated
> by all the elements $f(x_f)$. This ideal is proper: a finite relation
> $\sum g_i f_i(x_{f_i}) = 1$ could be evaluated in a finite extension containing
> a root $\alpha_i$ of each $f_i$, forcing $0 = 1$. A proper ideal lies in a
> maximal ideal $M$ (this is where Zorn's lemma enters), and
> $K_1 = F[\dots, x_f, \dots]/M$ is a field extending $F$ in which every $f$ over
> $F$ has a root, namely the class of $x_f$. Repeating the construction builds a
> chain $F \subseteq K_1 \subseteq K_2 \subseteq \cdots$ whose union $K$ is
> algebraically closed, since any polynomial over $K$ has its finitely many
> coefficients in some $K_N$ and thus a root in $K_{N+1}$. The elements of $K$
> algebraic over $F$ form an algebraic closure of $F$.[^df-prop30]

$$
% caption: Artin's tower: each field supplies roots for all polynomials over the
% one below, and the union of the chain is algebraically closed.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\node (F) at (0,0) {F};
\node (K1) at (0,1.3) {K1};
\node (K2) at (0,2.6) {K2};
\node (dots) at (0,3.6) {up};
\node[draw=acc, thick, fill=acc!8] (K) at (0,4.7) {K, closed};
\draw[thick] (F) -- (K1) -- (K2) -- (dots);
\draw[thick, acc] (dots) -- (K);
\node[font=\footnotesize, align=left, text=black] at (3.7,2.3)
  {each K adds a root for\\every polynomial over\\the one beneath it;\\the union closes up};
\end{tikzpicture}
$$

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.

[^df-thm25]: **Dummit & Foote**, §13.4, Theorem 25 — existence of a splitting field, by induction on degree adjoining one root at a time.
[^df-prop26]: **Dummit & Foote**, §13.4, Proposition 26 — a splitting field of a degree-$n$ polynomial has degree at most $n!$.
[^df-thm27]: **Dummit & Foote**, §13.4, Theorem 27 and Corollary 28 — extension of a base-field isomorphism to splitting fields, giving uniqueness up to isomorphism.
[^df-prop29]: **Dummit & Foote**, §13.4, Proposition 29 — an algebraic closure is itself algebraically closed.
[^df-prop30]: **Dummit & Foote**, §13.4, Propositions 30–31 and Corollary 32 — Artin's construction of an algebraically closed field, the algebraic closure inside it, and its uniqueness; $\mathbb{C}$ as the closure of its subfields.