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

5. Perfect fields 31has characteristic zero. But it can happen that L is a proper subfield ofΩ.Let K/k be an algebraic extension and L the maximal separable subfield.Consider the exponents of all elements in K. Let e be the maximumof these if it exists. we call e the exponent of the extension K/k. Itcan happen that e is finite but K/L is infinite.If K/k is a finite extension then K/L has degree p f so that the maximume of the exponents of elements of K exists. If e is the exponent ofK/k thene≤ fIt can happen that e< f . For instance let k have characteristic p0and letα∈k be not a p th power in k. Then k(α 1/p ) is of degree p overk. Letβin k be not a p th power in k. Then k(α 1/p ,β 1/p ) is of degree p 2over k,βk(α 1/p ) and for everyλ∈k(α 1/p ,β 1/p ),λ p ∈ k.We may for instance take k(x, y) to be the field of rational functionsof two variables and K = k(x 1/p , y 1/p ). Then (K : k(x, y))= p 2 andλ p ∈ k(x, y) for everyλ∈K.5 Perfect fieldsLet k be a field of characteristic p>0. LetΩbe its algebraic closure.Letω∈k. Then there is only one elementω ′ ∈Ω such thatω ′p =ω.We can therefore writeω 1/p without any ambiguity. Let k p−1 be the fieldgenerated inΩ/k by the p th roots of all elements of k. Similarly from 37k p−2 ,... Let ⋃K= k p−nn≥0Obviously K is a field; for ifα,β∈K,α,β∈k p−n for some large n.We denote K by k p−∞We shall study k p−∞ in relation to k andΩ. k p−∞ is called the rootfield of k.Letω∈k p−∞ . Thenω∈k p−∞ for some n so thatω p∞ ∈ k orωispurely inseparable. On the other hand letω∈Ω be purely inseparable.Thenω pn ∈ k for some n i.e.,ω∈k p−∞ ⊂ k p−∞ . Thus