In case the Galois group is cyclic as above, generated by formula_29 with formula_30 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 formula_31; then any distinct characters formula_32 are linearly independent in the "K"-vector space of mappings. We apply this to the Galois group automorphisms formula_33 thought of as mappings from the multiplicative group formula_34. Now formula_35as an "F"-vector space, so we may consider formula_36 as an element of the matrix algebra formula_37 since its powers formula_38 are linearly independent (over "K" and a fortiori over "F"), its minimal polynomial must have degree at least "n", i.e. it must be formula_39.