---
tag: 01441KDGBY
kind: statement
role: theorem
name: Existence of splitting fields
msc: [12F05]
uses: [0VG02M1FEZ, 02NZZ1J0FH, 0A7KJ68XHT]
sources:
  - url: https://en.wikipedia.org/wiki/Splitting_field
    locator: Constructing splitting fields
---

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ have degree $n \ge 1$.
Then $f$ has a [[splitting field|02NZZ1J0FH]] $L$ over $K$, and $[L : K] \le n!$.

## Remarks

The proof is by induction on $n$. [[Some extension $K_1$ contains a root $a_1$ of $f$|0VG02M1FEZ]], with $[K_1 : K] \le n$. Then $f = (x - a_1) g$ in $K_1[x]$, and a splitting field of $g$ over $K_1$ is a splitting field of $f$ over $K$.
The bound $n!$ follows from the [[tower law|0A7KJ68XHT]].
The bound is reached: the splitting field of $x^3 - 2$ over $\mathbb{Q}$ has degree $6 = 3!$.

## Explanations

### L1 human

You build a splitting field one root at a time.
Each step adds a root of a remaining factor, and the degree of the step is at most the degree of that factor.