Affine Algebraic Sets Review

We will now review some of the recent material regarding affine algebraic sets.

• On the Affine n-Space over a Field page we said that if $K$ is a field then the **Affine $n$-Space over $K$ is denoted by $\mathbb{A}^n(K) = K^n$ (where we sometimes denote it simply by $\mathbb{A}^n$ when the field $K$ is understood) and the points of this space are $n$-tuples $(x_1, x_2, ..., x_n)$ such that $x_1, x_2, ..., x_n \in K$.

• We then said that if $K$ is a field and $F \in K[x_1, x_2, ..., x_n]$ then a point $\mathbf{p} \in \mathbb{A}^n(K)$ is said to be a Zero or Root of $F$ if:

(1)

\begin{align} \quad F(\mathbf{p}) = 0 \end{align}

• On the Affine Algebraic Sets page we said that if $K$ is a field and $S \subseteq K[x_1, x_2, ..., x_n]$ then the Zero Locus of $S$ of the Vanishing Set of $S$ is the set:

(2)

\begin{align} \quad V(S) = \{ \mathbf{p} \in \mathbb{A}^n(K) : F(\mathbf{p}) = 0, \: \forall F \in S \} \end{align}

• That is, $V(S)$ is the set of all points in affine $n$-space which cause every polynomial in $S$ to vanish simultaneously.

• We then said that a subset $X \subseteq \mathbb{A}^n(K)$ is an Affine Algebraic Set if there exists an $S \subseteq K[x_1, x_2, ..., x_n]$ such that:

(3)

\begin{align} \quad X = V(S) \end{align}

• That is, the affine algebraic sets are precisely the vanishing sets of subsets of polynomials in $K[x_1, x_2, ..., x_n]$.

• On the Basic Properties of Affine Algebraic Sets page we listed a bunch of properties of affine algebraic sets which are summarized below.

Number	Property
1	If $S, T \subseteq K[x_1, x_2, ..., x_n]$ and $S \subseteq T$ then $V(S) \supseteq V(T)$.
2	The union of a finite collection of affine algebraic sets is an affine algebraic set.
3	The intersection of an arbitrary collection of affine algebraic sets is an affine algebraic set.
4	$V(0) = \mathbb{A}^{n}(K)$ and $V(1) = \emptyset$.