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

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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  .

Group representation point of view

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A field extension K / F 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. Representations of G over the field F can be viewed as left modules for the group algebra F[G]. Every homomorphism of left F[G]-modules   is of form   for some  . Since   is a linear basis of F[G] over F, it follows easily that   is bijective iff   generates a normal basis of K over F. The normal basis theorem therefore amounts to the statement saying that if K / F is finite Galois extension, then   as left  -module. In terms of representations of G over F, this means that K is isomorphic to the regular representation.

Case of finite fields

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For finite fields this can be stated as follows:[1] Let   denote the field of q elements, where q = pm is a prime power, and let   denote its extension field of degree n ≥ 1. Here the Galois group is   with   a cyclic group generated by the q-power Frobenius automorphism  with   Then there exists an element βK such that   is a basis of K over F.

Proof for finite fields

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In case the Galois group is cyclic as above, generated by   with   the normal basis theorem follows from two basic facts. The first is the linear independence of characters: a multiplicative character is a mapping χ from a group H to a field K satisfying  ; then any distinct characters   are linearly independent in the K-vector space of mappings. We apply this to the Galois group automorphisms   thought of as mappings from the multiplicative group  . Now  as an F-vector space, so we may consider   as an element of the matrix algebra Mn(F); since its powers   are linearly independent (over K and a fortiori over F), its minimal polynomial must have degree at least n, i.e. it must be  .

The second basic fact is the classification of finitely generated modules over a PID such as  . Every such module M can be represented as  , where   may be chosen so that they are monic polynomials or zero and   is a multiple of  .   is the monic polynomial of smallest degree annihilating the module, or zero if no such non-zero polynomial exists. In the first case  , in the second case  . In our case of cyclic G of size n generated by   we have an F-algebra isomorphism   where X corresponds to  , so every  -module may be viewed as an  -module with multiplication by X being multiplication by  . In case of K this means  , so the monic polynomial of smallest degree annihilating K is the minimal polynomial of  . Since K is a finite dimensional F-space, the representation above is possible with  . Since   we can only have  , and   as F[X]-modules. (Note this is an isomorphism of F-linear spaces, but not of rings or F-algebras.) This gives isomorphism of  -modules   that we talked about above, and under it the basis   on the right side corresponds to a normal basis   of K on the left.

Note that this proof would also apply in the case of a cyclic Kummer extension.

Example

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Consider the field   over  , with Frobenius automorphism  . The proof above clarifies the choice of normal bases in terms of the structure of K as a representation of G (or F[G]-module). The irreducible factorization   means we have a direct sum of F[G]-modules (by the Chinese remainder theorem):  The first component is just  , while the second is isomorphic as an F[G]-module to   under the action   (Thus   as F[G]-modules, but not as F-algebras.)

The elements   which can be used for a normal basis are precisely those outside either of the submodules, so that   and  . In terms of the G-orbits of K, which correspond to the irreducible factors of:   the elements of   are the roots of  , the nonzero elements of the submodule   are the roots of  , while the normal basis, which in this case is unique, is given by the roots of the remaining factor  .

By contrast, for the extension field   in which n = 4 is divisible by p = 2, we have the F[G]-module isomorphism   Here the operator   is not diagonalizable, the module L has nested submodules given by generalized eigenspaces of  , and the normal basis elements β are those outside the largest proper generalized eigenspace, the elements with  .

Application to cryptography

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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.

For example, in the field   above, we may represent elements as bit-strings:   where the coefficients are bits   Now we can square elements by doing a left circular shift,  , since squaring β4 gives β8 = β. This makes the normal basis especially attractive for cryptosystems that utilize frequent squaring.

Proof for the case of infinite fields

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Suppose   is a finite Galois extension of the infinite field F. Let [K : F] = n,  , where  . By the primitive element theorem there exists   such   and  . Let us write  .  's (monic) minimal polynomial f over K is the irreducible degree n polynomial given by the formula   Since f is separable (it has simple roots) we may define   In other words,   Note that   and   for  . Next, define an   matrix A of polynomials over K and a polynomial D by   Observe that  , where k is determined by  ; in particular   iff  . It follows that   is the permutation matrix corresponding to the permutation of G which sends each   to  . (We denote by   the matrix obtained by evaluating   at  .) Therefore,  . We see that D is a non-zero polynomial, and therefore it has only a finite number of roots. Since we assumed F is infinite, we can find   such that  . Define   We claim that   is a normal basis. We only have to show that   are linearly independent over F, so suppose   for some  . Applying the automorphism   yields   for all i. In other words,  . Since  , we conclude that  , which completes the proof.

It is tempting to take   because  . But this is impermissible because we used the fact that   to conclude that for any F-automorphism   and polynomial   over   the value of the polynomial   at a equals  .

Primitive normal basis

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A primitive normal basis of an extension of finite fields E / F is a normal basis for E / F that is generated by a primitive element of E, that is a generator of the multiplicative group K×. (Note that this is a more restrictive definition of primitive element than that mentioned above after the general normal basis theorem: one requires powers of the element to produce every non-zero element of K, not merely a basis.) Lenstra and Schoof (1987) proved that every finite field extension possesses a primitive normal basis, the case when F is a prime field having been settled by Harold Davenport.

Free elements

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If K / F is a Galois extension and x in K generates a normal basis over F, then x is free in K / F. If x has the property that for every subgroup H of the Galois group G, with fixed field KH, x is free for K / KH, then x is said to be completely free in K / F. Every Galois extension has a completely free element.[2]

See also

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References

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  1. ^ Nader H. Bshouty; Gadiel Seroussi (1989), Generalizations of the normal basis theorem of finite fields (PDF), p. 1; SIAM J. Discrete Math. 3 (1990), no. 3, 330–337.
  2. ^ Dirk Hachenberger, Completely free elements, in Cohen & Niederreiter (1996) pp. 97–107 Zbl 0864.11066