closed sets and zariski topology

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Lecture 1 Closed sets and the Zariski topology Let k be an infinite field (e.g. Q, R., C, or F q ), and let k[x1 , . . . xn ] be the polynomial ring in n indeterminants. We will often abbreviate k[x1 , . . . , xn ] by k[x]. We call an n-tuple α = (α1 , . . . , αn ) ∈ Zn+ a multi-index. For any multi- index α, define the monomial xα by xα := xα1 1 xα2 2 . . . xαnn . We also define the degree of xα to be deg(xα ) = αi . The point of assming P k is infinite is the next Proposition. Proposition 1.1. Let…

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