FoldUnfold Table of Contents The Hilbert Basis Theorem The Hilbert Basis Theorem Recall from the Noetherian Rings page that a ring $R$ is said to be a Noetherian ring if it satisfies the ascending chain condition, that is, for all ascending chains of ideals $I_1 \subseteq I_2 \subseteq ... \subseteq I_n \subseteq ...$ there exists an $N \in \mathbb{N}$ such that for all $m … [Read more...]

## Noetherian Rings

FoldUnfold Table of Contents Noetherian Rings Noetherian Rings Definition: A ring $R$ is said to be a Noetherian Ring if it satisfies the following condition (known as the Ascending Chain Condition): Every infinite ascending chain of ideals $I_1 \subseteq I_2 \subseteq ... \subseteq I_n \subseteq ...$ stabilizes, that is, there exists an $N \in \mathbb{N}$ such that for all … [Read more...]

## The Ideal of a Set of Points Review

FoldUnfold Table of Contents The Ideal of a Set of Points Review The Ideal of a Set of Points Review We will now review some of the recent material regarding the ideal of a set of points. On the [[[The Ideal of a Set of Points]] we said that if $K$ is a field and $X \subseteq \mathbb{A}^n(K)$ then the Ideal of $X$ is defined as the ideal: (1) \begin{align} \quad I(X) = \{ F … [Read more...]

## The Ideal of a Set of Points is a Radical Ideal

FoldUnfold Table of Contents The Ideal of a Set of Points is a Radical Ideal The Ideal of a Set of Points is a Radical Ideal Recall from the The Ideal of a Set of Points page that if $K$ is a field then the ideal of $X$ is defined as: (1) \begin{align} \quad I(X) = \{ F \in K[x_1, x_2, ..., x_n] : F(\mathbf{p}) = 0, \: \forall \mathbf{p} \in X \} \end{align} We will shortly … [Read more...]

## Basic Properties of the Ideal of a Set of Points

FoldUnfold Table of Contents Basic Properties of the Ideal of a Set of Points Basic Properties of the Ideal of a Set of Points Recall from The Ideal of a Set of Points page that if $K$ is a field and $X \subseteq \mathbb{A}^n(K)$ then the ideal of $X$ is defined as: (1) \begin{align} \quad I(X) = \{ F \in K[x_1, x_2, ..., x_n] : F(\mathbf{p}) = 0, \: \forall \mathbf{p} \in … [Read more...]

## The Ideal of a Set of Points

FoldUnfold Table of Contents The Ideal of a Set of Points The Ideal of a Set of Points Definition: Let $K$ be a field and let $X \subseteq \mathbb{A}^n(K)$. The Ideal of $X$ is defined as $I(X) = \{ F \in K[x_1, x_2, ..., x_n] : F(\mathbf{p}) = 0, \: \forall \mathbf{p} \in X \}$. We verify that $I(X)$ defined above is indeed an ideal. Theorem 1: Let $K$ be a field and let … [Read more...]

## Affine Algebraic Sets Review

FoldUnfold Table of Contents Affine Algebraic Sets Review 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 … [Read more...]

## Basic Properties of Affine Algebraic Sets

FoldUnfold Table of Contents Basic Properties of Affine Algebraic Sets Basic Properties of Affine Algebraic Sets Recall from the Affine Algebraic Sets page that if $K$ is a field and $S \subseteq K[x_1, x_2, ..., x_n]$ then the zero locus of $S$ is defined to be: (1) \begin{align} \quad V(S) = \{ \mathbf{p} \in \mathbb{A}^n(K) : F(\mathbf{p}) = 0, \: \forall F \in S \} … [Read more...]

## Affine Algebraic Sets

FoldUnfold Table of Contents Affine Algebraic Sets Affine Algebraic Sets Recall from the Affine n-Space over a Field page that if $K$ is a field then we denote the affine n-space over $K$ to be the set: (1) \begin{align} \quad \mathbb{A}^n(K) = \underbrace{K \times K \times ... \times K}_{n \: \mathrm{times}} = K^n \end{align} Furthermore, $K[x_1, x_2, ..., x_n]$ denotes … [Read more...]

## Affine n-Space over a Field

FoldUnfold Table of Contents Affine n-Space over a Field Affine n-Space over a Field Definition: Let $K$ be a field. The Affine $n$-Space over $K$ is the set $\mathbb{A}^n(K) = K^n$. The Points in $\mathbb{A}^n(K)$ are the elements of $\mathbb{A}^n(K)$. The notation "$K^n$" denotes the cartesian product $K^n = \underbrace{K \times K \times ... \times K}_{n \: … [Read more...]