FoldUnfold Table of Contents The Multiplicity of an Affine Plane Curve at a Point The Multiplicity of an Affine Plane Curve at a Point Definition: Let $K$ be a field and let $F \in K[x, y]$ be an affine plane curve. Let $F = F_m + F_{m+1} + … + F_n$ where each $F_i$ is […]

## Simple and Multiple Points of Affine Plane Curves

FoldUnfold Table of Contents Simple and Multiple Points of Affine Plane Curves Simple and Multiple Points of Affine Plane Curves Recall from the Affine Plane Curves page that we said that two polynomials $F, G \in K[x, y]$ are said to be equivalent if there exists a nonzero $\lambda \in K$ such that $F = […]

## Affine Plane Curves

FoldUnfold Table of Contents Affine Plane Curves Affine Plane Curves Definition: Let $K$ be a field. Two polynomials $F, G \in K[x, y]$ are said to be Equivalent if there exists a nonzero $\lambda \in K$ such that $F = \lambda G$. This forms an equivalence relation on the set of polynomials in $K[x, y]$. […]

## Polynomial Forms

FoldUnfold Table of Contents Polynomial Forms Polynomial Forms Definition: Let $R$ be a ring. A polynomial $F \in R[x_1, x_2, …, x_n]$ is said to be a Form of Degree $d$ if every term in $F$ is of degree $d$. For example, the following polynomials are forms of degree $2$, $3$, and $4$ respectively: (1) […]

## Discrete Valuation Rings

FoldUnfold Table of Contents Discrete Valuation Rings Discrete Valuation Rings Definition: A Discrete Valuation Ring (DVR) is an integral domain $R$ with the following properties:1) $R$ is a Noetherian ring.2) $R$ is a local ring.3) The unique maximal ideal of $R$ is a principal ideal. Recall that an ideal $I$ of a ring $R$ is […]

## Local Rings

FoldUnfold Table of Contents Local Rings Local Rings Definition: A Local Ring is a ring $R$ that has a unique maximal ideal. Recall that a proper ideal $I$ of a ring $R$ is a maximal ideal if there exists no other proper ideals of $R$ containing $I$. In general, maximal ideals need not be unique. […]

## The Rational Function Field of an Affine Variety

FoldUnfold Table of Contents The Rational Function Field of an Affine Variety The Rational Function Field of an Affine Variety Definition: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ be a nonempty affine variety. The Rational Function Field of $V$ is defined as $K(V) = \left \{ \frac{f}{g} : f, g \in \Gamma(V), […]

FoldUnfold Table of Contents Recall from The Coordinate Ring of an Affine Variety page that if $K$ is a field and $V \subseteq \mathbb{A}^n(K)$ is a nonempty affine variety then the coordinate ring of $V$ is defined as: (1) \begin{align} \quad \Gamma(V) = K[x_1, x_2, …, x_n]/I(V) \end{align} We noted that since $V$ is an […]

## The Coordinate Ring of an Affine Variety

FoldUnfold Table of Contents The Coordinate Ring of an Affine Variety The Coordinate Ring of an Affine Variety Recall from the Affine Varieties page that an affine variety is simply an irreducible affine algebraic set. When we have a nonempty affine variety $V$ we can consider what is called the coordinate ring of $V$ which […]

## Affine Varieties

FoldUnfold Table of Contents Affine Varieties Affine Varieties Definition: Let $K$ be a field and let $X \subseteq \mathbb{A}^n(K)$ be an affine algebraic set. Then $X$ is said to be Reducible if there exists affine algebraic sets $X_1$ and $X_2$ where $X_1, X_2 \neq \emptyset$ and $X_1, X_2 \neq X$ and such that $X = […]

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