---
tag: 069A3BC3EP
kind: statement
role: theorem
name: Uniqueness of splitting fields
msc: [12F05]
uses: [02NZZ1J0FH, 0ZN39QDTP9]
sources:
  - url: https://en.wikipedia.org/wiki/Splitting_field
    locator: Constructing splitting fields
---

## Statement

Let $K$ be a [[field|0QA8CHH8MG]], let $f \in K[x]$ be nonzero, and let $L$ and $L'$ be [[splitting fields|02NZZ1J0FH]] of $f$ over $K$.
Then there is a ring isomorphism $\sigma \colon L \to L'$ with $\sigma(c) = c$ for every $c \in K$.

## Remarks

The proof is by induction on $[L : K]$. It extends an isomorphism one root at a time, with the [[structure of simple extensions|0ZN39QDTP9]].
The isomorphism $\sigma$ is usually not unique. The isomorphisms from $L$ to itself that fix $K$ form the [[Galois group|0XVR72M51M]] of $f$.
So one speaks of "the" splitting field of $f$ over $K$.
The choice of the irreducible factors in the construction does not change the result, up to isomorphism.

## Explanations

### L1 human

Two splitting fields of the same polynomial are the same field with different names for the elements.
The proof extends a matching of the base field, one root at a time.