In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis 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, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory.

Normal basis theorem

. The classical normal basis theorem, considered as a vector space.

A normal basis contrasts with a primitive element.

Group representation point of view

A field extension with Galois group G can be naturally viewed as a representation of the group G over the field F in which each automorphism is represented by itself; thus K is also a left module for the group algebra F[G]. Every homomorphism of left F[G is a linear basis of F[G] over F generates a normal basis of K over F.

The normal basis theorem therefore amounts to the statement saying that if -module: K is isomorphic to the regular representation generates a normal basis iff, when considering K as a G does not lie in any proper subrepresentation.

Case of finite fields

For finite fields this can be stated as follows: denote the field of q elements, where is a prime power denote its extension field of degree a cyclic group generated by the q-power Frobenius automorphism Then there exists an element is a basis of K over F.

Proof for finite fields

the normal basis theorem follows from two basic facts. The first is the linear independence of characters: a multiplicative character are linearly independent in the as an F are linearly independent over and a fortiori over , its minimal polynomial must have degree at least n.

The second basic fact is the classification of modules over a PID of finite dimension over F are monic polynomials of order n, we have an : annihilating as -modules (but this is not-modules , of on the left.

Note that this proof would also apply in characteristic zero for a cyclic Kummer extension.

Example

. The proof above clarifies the choice of normal bases in terms of the structure of K as a representation of G (or means we have a direct sum of F[G]-modules (by the Chinese remainder theorem, while the second is isomorphic as an as -modules, but not as rings.)

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in which is divisible by , we have the is not diagonalizable has nested submodules given by generalized eigenspaces, and the normal basis elements β.

Application to cryptography

The normal basis is frequently used in cryptographic applications based on the discrete logarithm problem, such as elliptic curve cryptography, since arithmetic using a normal basis is typically more computationally efficient than using other bases.

Now we can square elements by doing a left circular shift, since squaring β4 gives . This makes the normal basis especially attractive for cryptosystems that utilize frequent squaring.

Primitive normal basis

A primitive normal basis is a normal basis that is generated by a primitive element. (Note that this is a stronger definition of primitive element than mentioned above: one requires powers of the element to produce every non-zero element of , not merely a basis.) Lenstra and Schoof (1987) proved that every extension of finite fields possesses a primitive normal basis, the case when is a prime field having been settled by Harold Davenport.

Proof for the case of infinite fields

is a finite Galois extension of the infinite field F. By the primitive element theorem. The minimal polynomial f over Ka degree n. Since f is separable (having simple roots) we may define the Lagrange interpolation polynomials, where k is the permutation matrix corresponding to the permutation of G is a non-zero polynomial, it has only a finite number of roots; and since we assumed F are linearly independent over F, which completes the proof.

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Free elements

If is a Galois extension and x in K generates a normal basis over F, then x is free in . If x has the property that for every subgroup H of the Galois group G, with fixed field KH, x is free for , then x is said to be completely free in . Every Galois extension has a completely free element.

See also

• Dual basis in a field extension

• Polynomial basis

• Zech's logarithm

References