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 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 -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 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