This preview shows pages 1–2. 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: Solution to exercise 2.2?? David Krumm Let ( K,   ) be a normed field. Fix a positive integer n and let P n denote the set of all degree n polynomials over K which have distinct roots in K . There is a natural injection P n , K n +1 given by n i =0 a i x i 7 ( a ,...,a n ). We denote by D ( n ) the image of P n under this map. We will denote by A n +1 the set K n +1 endowed with the Zariski topology, and we will reserve the notation K n +1 for the same set endowed with the product topology. (a) The set D ( n ) is open in A n +1 . If p ( x ) K [ x ] is any polynomial, we may consider its discriminant ( p ), which is an element of K having the property that ( p ) = 0 if and only if p has a repeated root. Moreover, there is an explicit formula for ( p ) as a polynomial in the coefficients of p . We therefore have a polynomial K [ t ,...,t n ] such that A n +1 \ D ( n ) is the union of the zero set of with the zero set of the last coordinate function. Since this union is clearly a closed set, it follows thatcoordinate function....
View Full
Document
 Fall '11
 Staff
 Math, Number Theory, Polynomials

Click to edit the document details