Abstract
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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Lang, S. (2002). Algebraic Spaces. In: Algebra. Graduate Texts in Mathematics, vol 211. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0041-0_9
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