Jim8410 - SEPARABLE EXTENSIONS OF DEGREE p IN...

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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 Artin-Schreier 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 well-known 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)....
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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.

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Jim8410 - SEPARABLE EXTENSIONS OF DEGREE p IN...

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