Linear Independence of Characters

Example 1.1. A field homomorphism K → F is a character by restricting it to the nonzero elements of K (that is, using G = K×) and ignoring the additive aspect of a field homomorphism. In particular, when L/K is a field extension any element of Aut(L/K) is a field homomorphism L→ L and therefore is a character of L× with values in L×. Example 1.2. For any α ∈ F×, the map Z→ F× by k 7→ αk is a character of Z. Characters G→ F× can be regarded as special functions G→ F and then can be added, but the sum is no longer a character (since the sum of multiplicative maps is usually not multiplicative, and could even take the value 0). The sum is just a function G → F . The functions G → F form a vector space under addition and F -scaling. We will prove that different characters G → F× are linearly independent as functions G → F . Then we turn to three very important applications of this linear independence: • The normal basis theorem. • Hilbert’s Theorem 90 for cyclic Galois extensions. • Some basic ideas in Kummer theory and Artin-Schreier theory. While we will use Galois theory to prove results about characters, in [3] and [8] linear independence of characters is used to prove the Galois correspondence. That approach to Galois theory is due to Artin [1], who I think wanted to avoid the primitive element theorem.