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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....
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