---
tag: PXY1WYNRD4
kind: statement
role: theorem
name: Normal basis theorem
msc: [12F10, 11R33]
uses: [0DS2F7GK05, 0XVR72M51M, 0VTQTPQGAV, N43G50YHG2]
about: [0QMZAQ6QQ4]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_basis
---

## Statement

Let $L/K$ be a finite [[Galois extension|0DS2F7GK05]] with [[Galois group|0XVR72M51M]] $G = \Gal(L/K)$.
Then there is an element $\alpha \in L$ such that the family $(\sigma(\alpha))_{\sigma \in G}$ is a [[basis|0VTQTPQGAV]] of $L$ as a $K$-vector space.

## Remarks

Such a basis is called a normal basis, and $\alpha$ is a normal basis generator.
Equivalently, $L$ is isomorphic to the group algebra $K[G]$ as a module over $K[G]$.

When $K$ is infinite, write $G = \set{\sigma_1, \ldots, \sigma_n}$ and fix a basis $b_1, \ldots, b_n$ of $L$.
The family $(\sigma_i(\alpha))$ is a basis exactly when $\det\left(\sigma_i^{-1}\sigma_j(\alpha)\right) \ne 0$; writing $\alpha = \sum x_k b_k$ makes this determinant a polynomial in $x_1, \ldots, x_n$.
An argument in the spirit of [[Dedekind's lemma|N43G50YHG2]], using that $K$ is infinite, shows that this polynomial is not zero.
A nonzero polynomial over an infinite field has a non-root in $K^n$.
When $K$ is finite, $G$ is cyclic, generated by the Frobenius $\varphi$.
Dedekind's lemma shows that the minimal polynomial of the $K$-linear map $\varphi$ is $x^n - 1$, and the theory of cyclic modules then gives the normal basis.

Example: $L = \Q(i)$ over $K = \Q$, with $G = \set{\id, \sigma}$ and $\sigma(i) = -i$.
For $\alpha = 1 + i$, the family $1 + i, 1 - i$ is a basis of $\Q(i)$ over $\Q$.
The choices $\alpha = 1$ and $\alpha = i$ fail: they give $1, 1$ and $i, -i$, which are linearly dependent.

## Explanations

### L1 human

A Galois extension has a group of symmetries.
The normal basis theorem says that one well-chosen element, together with all its images under the symmetries, already forms a basis.
So the whole field is spanned by the conjugates of a single element.

### L2 human

In coordinates of a normal basis, the Galois group acts by permuting the coordinates, which makes many computations simple.
Over finite fields, normal bases make the Frobenius map $x \mapsto x^q$ a cyclic shift of coordinates, which is used in computer arithmetic.
The theorem says that $L$ is a free $K[G]$-module of rank $1$.
Whether the ring of integers of a number field is free over $\Z[G]$ is a much subtler question of the theory of Galois modules.