Lectures on the Algebraic Theory of Fields - Tata Institute of ...

5. Perfect fields 335) Any algebraic extension of a perfect field is perfect.For let K/k be algebraic and k be perfect. Ifαis inseparable overK, then it is already so over k.An example of an imperfect field is the field of rational functionsof one variable x over a finite field k. For if k has characteristicp, then x 1/p k(x) and k(x 1/p ) is a purely inseparable extensionover k(x).Note 1. Ifα∈Ω is inseparable over k, it is not true that it is inseparableover every intermediary field, whereas this is true ifαis separable. 39Note 2. If K/k is algebraic and K∩k p−∞ contains k properly then K is aninseparable extension. But the converse of this is not true, that is, if K/kis an inseparable extension, it can happen that there are no elements inK which are purely inseparable over k. We give to this end the followingexample due to Bourbaki.Let k be a field of characteristic p>2 and let f (x) by in irreduciblepolynomialf (x)= x n + a 1 x n−1 +···+a nin k[x]. Ifα 1 ,...,α t are the distinct roots of f (x) inΩthenf (x)= { (x−α 1 )...(x−α t ) } p e e≥1.where n=t.· p e . Putφ(x)= f (x p ). ThenIfβ i =α 1/pithenφ(x)= { (x p −α 1 )...(x p −α t )} peφ(x)= { (x−β 1 )...(x−β t ) } p e+1andβ,...,β t are distinct sinceα 1 ...α t are distinct. Supposeφ(x) isreducible in k[x] and letψ(x) be an irreducible factor ofφ(x) in k[x].Thenψ(x)= { (x−β 1 )···(x−β t ) } p µ