Let $k$ be a field. Affine algebraic geometry begins with a deceptively simple ambient space: $k^n$. At first it looks like ordinary coordinate space, but its real role is not metric or Euclidean. It is the place where formal polynomial expressions can be evaluated at points, where equations such as $x_1x_2-1=0$ cut out geometric objects, and where algebra can read geometry through the ring $k[x_1,\ldots,x_n]$.

[example: A Curve as a Polynomial Equation] Let $k$ be a field and consider $f=x_1x_2-1 \in k[x_1,x_2]$. For a point $a=(a_1,a_2)\in k^2$, the coordinate functions satisfy $x_1(a)=a_1$ and $x_2(a)=a_2$, so evaluating $f$ at $a$ gives \begin{align*} f(a)=x_1(a)x_2(a)-1=a_1a_2-1. \end{align*} Therefore the zero set of $f$ in $\mathbb{A}^2_k=k^2$ is \begin{align*} V(f)=\{(a_1,a_2)\in k^2 : f(a_1,a_2)=0\}=\{(a_1,a_2)\in k^2 : a_1a_2-1=0\}. \end{align*} Equivalently, a point $(a_1,a_2)$ lies on this set exactly when $a_1a_2=1$. This is not being introduced as a graph in the calculus sense; it is the solution set of one polynomial equation inside affine space. The ambient field is part of the data: the same formal expression $x_1x_2-1$ may have different sets of solutions when evaluated on $\mathbb{R}^2$, $\mathbb{C}^2$, or $k^2$ for a finite field $k$. [/example]

The same word "affine" also appears in affine geometry, where the emphasis is on points, translations, lines, and the absence of a distinguished origin. That viewpoint is useful, and this page returns to it later. The graph-context object for algebraic geometry, however, is the algebraic affine space: the set $k^n$ together with the formal coordinate algebra whose elements are evaluated on points.

## Definition

The first definition packages the ambient object for polynomial equations. The set of points alone is not enough for algebraic geometry: over finite fields, two different formal polynomials may give the same set-theoretic function on $k^n$. To avoid losing algebraic information, affine space remembers the coordinate functions and the formal polynomial algebra that supplies expressions to be evaluated.

[definition: Affine Space] Let $k$ be a field and let $n \in \mathbb{N}$. The $n$-dimensional affine space over $k$ is the set of points \begin{align*} \mathbb{A}^n_k=k^n \end{align*} equipped with its coordinate functions $x_1,\ldots,x_n:\mathbb{A}^n_k \to k$ and the formal coordinate algebra $k[x_1,\ldots,x_n]$ whose elements are evaluated on $\mathbb{A}^n_k$. [/definition]

This definition turns a point $a=(a_1,\ldots,a_n)$ into something polynomials can evaluate on. The notation $\mathbb{A}^n_k$ reminds us that the field is part of the data: the same equations may have different solution sets over $\mathbb{R}$, $\mathbb{C}$, or a finite field.

The next task is to make the variables $x_i$ into actual operations on points. Without this step, the symbol $x_i$ would only be a formal indeterminate in a ring; the coordinate function is what lets an equation read the $i$-th entry of a point in $\mathbb{A}^n_k$.

[definition: Coordinate Function on Affine Space] For $\mathbb{A}^n_k=k^n$, the $i$-th coordinate function is the map \begin{align*} x_i:\mathbb{A}^n_k \to k. \end{align*} It is defined by \begin{align*} x_i(a_1,\ldots,a_n)=a_i \end{align*} for $1 \le i \le n$. [/definition]

Once the coordinate functions are available, the next question is which scalar-valued functions count as algebraic. Algebraic geometry does not permit arbitrary maps $\mathbb{A}^n_k \to k$; it permits the functions obtained by evaluating formal expressions built from the coordinates by addition and multiplication.

[definition: Polynomial Function on Affine Space] Let $\mathbb{A}^n_k$ be affine space over $k$. A polynomial function on $\mathbb{A}^n_k$ is a function $F:\mathbb{A}^n_k \to k$ obtained by evaluating a polynomial $f \in k[x_1,\ldots,x_n]$: \begin{align*} F(a_1,\ldots,a_n)=f(a_1,\ldots,a_n). \end{align*} [/definition]

This definition is pointwise: it describes the induced function on $k$-points. The formal polynomial that produces the function remains part of the algebraic background, because pointwise equality can collapse distinct expressions over small fields.

[example: Different Polynomials, Same Function over a Finite Field] Over $k=\mathbb{F}_p$, every element of $k$ has the form $n\cdot 1$ for some $n\in\{0,\ldots,p-1\}$. Since $k$ has characteristic $p$, the binomial coefficients $\binom{p}{j}$ vanish in $k$ for $1\le j\le p-1$, so \begin{align*} (u+v)^p=u^p+v^p \end{align*} for all $u,v\in k$. Starting from $0^p=0$, induction gives \begin{align*} ((n+1)\cdot 1)^p=(n\cdot 1+1)^p=(n\cdot 1)^p+1^p=n\cdot 1+1=(n+1)\cdot 1. \end{align*} Thus $a^p=a$ for every $a\in\mathbb{F}_p$, and the two polynomial functions $\mathbb{A}^1_k\to k$ induced by $x^p$ and $x$ agree pointwise: \begin{align*} x^p(a)=a^p=a=x(a). \end{align*} As formal polynomials in $k[x]$, however, they are not equal, because \begin{align*} x^p-x\ne 0 \end{align*} has nonzero coefficient $1$ on the monomial $x^p$ and nonzero coefficient $-1$ on the monomial $x$. The point is that equality as functions on $k$-points can identify distinct formal polynomial expressions, so algebraic geometry keeps track of coordinate rings and not only set-theoretic functions. [/example]

The finite-field example shows why the formal algebra cannot be replaced by raw functions too soon. The next thing the theory needs is a disciplined way to pass from equations to geometry: given a family of formal polynomials, we want the set of points on which all of them vanish simultaneously. This construction is the basic bridge from algebra to subsets of affine space, and it is flexible enough to handle one equation, many equations, and even an infinite family of equations at once.

[definition: Affine Algebraic Set] Let $S \subset k[x_1,\ldots,x_n]$. The affine algebraic set cut out by $S$ is \begin{align*} V(S)=\{a \in \mathbb{A}^n_k : f(a)=0 \text{ for every } f \in S\}. \end{align*} [/definition]

Algebraic sets turn systems of polynomial equations into geometry. The notation $V(S)$ records the common vanishing locus of the equations in $S$, but the word "variety" is less uniform across the literature. We need a local convention before using it, because some authors reserve "affine variety" for irreducible algebraic sets while others use it for any affine algebraic set over an [algebraically closed field](/page/Algebraically%20Closed%20Field).

[definition: Affine Variety] In this page, an affine variety over $k$ is an affine algebraic set $X\subset \mathbb{A}^n_k$. [/definition]

This convention keeps the focus on the ambient affine-space construction. Irreducibility, reducedness, and scheme structure refine the notion later; the next example shows why that broad convention includes geometrically meaningful reducible sets rather than excluding them at the first pass.

[example: Coordinate Axes as an Algebraic Set] For $a=(a_1,a_2)\in \mathbb{A}^2_k=k^2$, evaluating the polynomial $x_1x_2$ gives \begin{align*} (x_1x_2)(a)=x_1(a)x_2(a)=a_1a_2. \end{align*} Thus \begin{align*} V(x_1x_2)=\{(a_1,a_2)\in k^2:a_1a_2=0\}. \end{align*} Because $k$ is a field, $a_1a_2=0$ is equivalent to $a_1=0$ or $a_2=0$: if $a_1=0$ or $a_2=0$, then $a_1a_2=0$; conversely, if $a_1a_2=0$ and $a_1\ne 0$, then $a_1^{-1}$ exists and \begin{align*} a_2=1a_2=(a_1^{-1}a_1)a_2=a_1^{-1}(a_1a_2)=a_1^{-1}0=0. \end{align*} Therefore \begin{align*} V(x_1x_2)=\{(a_1,a_2)\in k^2:a_1=0\text{ or }a_2=0\}. \end{align*} The condition $a_2=0$ is the $x_1$-axis, and the condition $a_1=0$ is the $x_2$-axis, so one factored equation cuts out the union of the two coordinate axes. This is why algebraic sets naturally include reducible objects: the factorization $x_1x_2$ records two components in the geometry. [/example]

The coordinate-axes example shows that geometry can remember algebraic features such as factorization and components. To go back in the other direction, from a subset of affine space to algebra, we need to ask which polynomial expressions cannot be detected on that subset because they vanish at every one of its points. Collecting exactly those expressions produces an ideal, and that ideal is the algebraic shadow of the point set.