This chapter gives the basic results concerning solutions of polynomial equations in several variables over a field k. First it will be proved that if such equations have a common zero in some field, then they have a common zero in the algebraic closure of k, and such a zero can be obtained by the process known as specialization. However, it is useful to deal with transcendental extensions of k as well. Indeed, if p is a prime ideal in k[X] = k[X 1, …, X n ], then k[X]/p is a finitely generated ring over k, and the images x i of X t in this ring may be transcendental over k, so we are led to consider such rings.
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