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.
KeywordsProjective Space Generic Point Prime Ideal Commutative Ring Polynomial Ring
Unable to display preview. Download preview PDF.
- [Br 87]
- [Br 88]
- [Br 89]D. Brownawell, Applications of Cayley-Chow forms, Springer Lecture Notes 1380: Number Theory, Ulm 1987, H. P. Schlickewei and E. Wirsing (eds.), pp. 1-18Google Scholar
- [Ko 88]