In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically the
algebraic theory of
fields, a normal basis is a special kind of
basis for
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
s of finite degree, characterised as forming a single
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
for the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, the study of the more refined question of the existence of a
normal integral basis is part of
Galois module theory.
Normal basis theorem
Let
be a Galois extension with Galois group
. The classical normal basis theorem states that there is an element
such that
forms a basis of ''K'', considered as a vector space over ''F''. That is, any element
can be written uniquely as
for some elements
A normal basis contrasts with a
primitive element basis of the form
, where
is an element whose minimal polynomial has degree