632HW2 - Math. 632. Homework 2 1. Let k = F ( t ) be a...

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Unformatted text preview: Math. 632. Homework 2 1. Let k = F ( t ) be a purely transcendental extension of a field F and K be an extension of k obtained by adjoining a p-th root of t . Compute the Galois group scheme of the extension K/k . 2. Let F be a functor from the category of algebras over a ring R to the category of sets. Assume F is representable by an affine scheme of finite type over R . Show that the functor F defined by F ( K ) = F ( K [[ T ]]) (check that this defines a functor) is representable by a scheme. 3. Show that X = Spec k [ X, Y ] / ( X, Y 3 ) is contained in a closed reduced subscheme of A 2 k of the form Spec( k [ X, Y ] / ( f ( X, Y )), where f ( x, y ) = 0 is a nonsingular curve. Show that for X = Spec k [ X, Y ] / ( X 2 , Y 2 , XY ) this is not true. 4. Let k/F be a finite extension of fields. consider the functor which assigns to a F-algebra K the group of invertible elements of k ⊗ F K . Show that this functor is representable by an affine group scheme. Find it explicitly in the casescheme....
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This note was uploaded on 02/24/2012 for the course MATH 632 taught by Professor Igordolgache during the Fall '08 term at University of Michigan-Dearborn.

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