The second basic fact is the classification of finitely generated modules over a PID such as formula_40, where formula_41 may be chosen so that they are monic polynomials or zero and formula_42 is a multiple of formula_41. formula_44 is the monic polynomial of smallest degree annihilating the module, or zero if no such non-zero polynomial exists. In the first case formula_45, in the second case formula_46. In our case of cyclic "G" of size "n" generated by formula_29 we have an "F"-algebra isomorphism formula_48 where "X" corresponds to formula_49, so every formula_50-module with multiplication by "X" being multiplication by formula_51. In case of "K" this means formula_52, so the monic polynomial of smallest degree annihilating "K" is the minimal polynomial of formula_29. Since "K" is a finite dimensional "F"-space, the representation above is possible with formula_54. Since formula_55 we can only have formula_56, and formula_57-modules. (Note this is an isomorphism of "F"-linear spaces, but "not" of rings or "F"-algebras!) This gives isomorphism of formula_58 that we talked about above, and under it the basis formula_59 on the right side corresponds to a normal basis formula_60 of "K" on the left.