Trace and Norm

Let L/K be a finite extension of fields, with n = [L : K]. We will associate to this extension two important functions L→ K, called the trace and the norm. For each α ∈ L, let mα : L → L be multiplication by α: mα(x) = αx for x ∈ L. Each mα is a K-linear map from L to L, so choosing a K-basis of L lets us write mα as an n×n matrix, which will be denoted [mα], or [mα]L/K if we need to emphasize the field extension that is involved. (We need to put an ordering on the basis to get a matrix, but we will often just refer to picking a basis and list it in a definite way instead of saying “pick an ordered basis”.)