over the value of the polynomial at equals . Primitive normal basis 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 . (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 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. See also Dual basis in a field extension Polynomial basis Zech's logarithm References Linear algebra Field (mathematics) Abstract algebra Cryptography