MA3412S2_Hil2014.pdf

Proposition 234 the ring r x 1 x 2 x n of polynomials

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Proposition 2.34 The ring R [ x 1 , x 2 , . . . , x n ] of polynomials in n indepen- dent indeterminates x 1 , x 2 , . . . , x n with coefficients in an integral domain R is itself an integral domain. Proof The result in the case n = 1 follows from the fact that the product of the leading coefficients of two non-zero polynomials f ( x 1 ) and g ( x 1 ) is a non-zero element of the integral domain R , and is thus the leading coefficient of the product polynomial f ( x 1 ) g ( x 1 ). The result when n > 1 then follows by induction on the number n of indeterminates in view of the fact that R [ x 1 , x 2 , . . . , x n ] = R [ x 1 , . . . , x n - 1 ][ x n ] for all n > 1. Proposition 2.35 The ring R [ x 1 , x 2 , . . . , x n ] of polynomials in n indepen- dent indeterminates x 1 , x 2 , . . . , x n with coefficients in a unique factorization domain R is itself a unique factorization domain. Proof The result for n = 1 was proved as Proposition 2.33. The result for n > 1 then follows by induction on the number n of indeterminates in view of the fact that R [ x 1 , x 2 , . . . , x n ] = R [ x 1 , . . . , x n - 1 ][ x n ] for all n > 1. 37

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Corollary 2.36 The ring K [ x 1 , x 2 , . . . , x n ] of polynomials in n independent indeterminates x 1 , x 2 , . . . , x n with coefficients in a field K is a unique factor- ization domain. Corollary 2.37 The ring Z [ x 1 , x 2 , . . . , x n ] of polynomials in n independent indeterminates x 1 , x 2 , . . . , x n with integer coefficients is a unique factoriza- tion domain. Example Let K be a field, and let K [ x, y ] be the ring of polynomials in two independent interminates x and y . Any polynomial f ( x, y ) in x and y with coefficients in K can be represented in the form g 0 ( x ) + d j =1 g j ( x ) y j , where g 0 ( x ) , g 1 ( x ) , . . . , g d ( x ) are polynomials in the indeterminate x with coefficients in K . These polynomials g 0 , g 1 , . . . , g d are uniquely determined by the polynomial f ( x ). There is thus a well-defined function ϕ : K [ x, y ] K [ x ], where ϕ g 0 ( x ) + d X j =1 g j ( x ) y j ! = g 0 ( x ) . for all g 0 , g 1 , . . . , g d K [ x ]. Moreover ϕ ( f )( x ) = f ( x, 0 K ) for all f K [ x, y ], where f ( x, 0 K ) denotes the polynomial in the indeterminate x obtained by substituting the zero element 0 K of the field K for the indeterminate y . The function ϕ : K [ x, y ] K [ x ] is a surjective ring homomorphism, and its kernel is the ideal P of K [ x, y ] generated by the polynomial y . Now K [ x, y ] /P = K [ x ], and K [ x ] is an integral domain. It follows that the principal ideal P of K [ x, y ] generated by the polynomial y is a prime ideal of K [ x, y ] (see Lemma 2.14). The function ε : K [ x, y ] K that maps a polynomial f ( x, y ) to its value f (0 K , 0 K ) at (0 K , 0 K ) is also a surjective ring homomorphism. It satisfies ε d x X j =0 d y X k =0 a j,k x j y k ! = a 0 , 0 for all coefficients a j,k , where a j,k K for j = 0 , 1 , . . . , d x and k = 0 , 1 , . . . , d y . The kernel of this homomorphism is the ideal M generated by the polyno- mials x and y . It follows that K [ x, y ] /M is a field isomorphic to the field K of coefficients, and thus M is a maximal ideal of K [ x, y ] (see Lemma 2.13).
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