In
mathematics, specifically the
algebraic theory of
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
, a normal basis is a special kind of
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
for
Galois extensions of finite degree, characterised as forming a single
orbit for the
Galois group. 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