This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: SEPARABLE EXTENSIONS OF DEGREE p IN CHARACTERISTIC p ; FAILURE OF HERMITE’S THEOREM IN CHARACTERISTIC p JIM STANKEWICZ 1. Separable Field Extensions of degree p Exercise: Let K be a field of characteristic p and ℘ p : K → K by x 7→ x p x be the ArtinSchreier isogeny. It is a group homomorphism with kernel ( F p , +). The separable degree p extensions of K are in some way in bijection with K/℘ p ( K ). 1.1. Solution 1 (Cohomology): Note first that ℘ p : K → K is onto, because if a ∈ K , we can find x ∈ K such that x p x = a because t p t a is an algebraic equation. Moreover the same is true replacing K with K sep since t p t a is a separable polynomial because its derivative over K is 1 6 = 0. In either case, the kernel is F p ⊂ K so we have the short exact sequence → F p → K sep ℘ p → K sep → . Clearly each of these is a G K = Gal( K sep /K )module, with F p possessing the trivial action since it is contained in K . (Thus, as a Galois module, F p ∼ = Z /p Z .) Hence H 1 ( G K , Z /p Z ) = Hom( G K , Z /p Z ). The natural thing to do then is to take the long exact sequence of cohomology: → Z /p Z → K ℘ p → K → H 1 ( G K , Z /p Z ) → H 1 ( G K ,K sep ) → .. . By the additive version of Hilbert’s Theorem 90, that H 1 ( G K ,K sep ) = 0, we have that K/℘ p ( K ) ∼ = H 1 ( G K , Z /p Z ) = hom( G K , Z /p Z ). Now recall that if we restrict our attention to separable degree p extensions which are actually Galois , these are in bijection with the open normal subgroups H of G K with quotient Z /p Z . Each of these gives a nonzero homomorphism r : G K → Z /p Z just by modding out by H . Now given homomorphisms φ,ψ : G K → Z /p Z with the same kernel H , we let g 1 ,. .. ,g p be a complete set of coset representatives for G K /H . We can define an automorphism α of G K /H by α ( φ ( g i )) = ψ ( g i ) since φ,ψ are both onto. Likewise if β is an automorphism of Z /p Z and φ : G K → Z /p Z is an onto homomorphism then ψ = β ◦ φ is a homomorphism with ker φ = ker ψ . Therefore the open normal sub groups of G K with quotient Z /p Z are in bijection with the nonconstant(equivalently onto) homomorphisms G K → Z /p Z modulo the automorphisms of Z /p Z , which are of course ( Z /p Z ) × . Important Note It’s crucial to mod out by this action as we can see in the case that K = Z /p Z . In this case ℘ p is identically zero, so K/℘ p ( K ) = K = Z /p Z , but it is wellknown that Z /p Z has exactly one degree p field extension. 1 2 JIM STANKEWICZ Therefore { Z /p Z cyclic extensions of K } ↔ ( Z /p Z ) × \ hom( G K , Z /p Z ) { } ∼ = ( Z /p Z ) × \ ( K/℘ p ( K ) { } ) Note that we use “double coset” notation even though K is abelian because ℘ p ( K ) is a normal subgroup of K while Z /p Z × acts on K by left multiplication (as we will see when we do this in a more concrete setting)....
View
Full
Document
This note was uploaded on 10/26/2011 for the course MATH 8410 taught by Professor Staff during the Fall '11 term at University of Georgia Athens.
 Fall '11
 Staff
 Number Theory

Click to edit the document details