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 $K^n = \underbrace{K \times K \times ... \times K}_{n \: \mathrm{times}}$. Therefore, points $\mathbb{A}^n(K)$ are $n$-tuples $(k_1, k_2, ..., k_n)$ where $k_1, k_2, ..., k_n \in K$.

If there is no ambiguity, the notation we may denote the affine $n$-space over $K$ by simply "$\mathbb{A}^n$" provided that the underlying field $K$ is understood.

For example, if $K = \mathbb{R}$ then $\mathbb{A}^2(\mathbb{R}) = \mathbb{R}^2$ is the affine $2$-space over $\mathbb{R}$ and the points in this space are of the form $(x, y)$ where $x, y \in \mathbb{R}$.

Now if $K$ is a field then $K$ is also a ring. So we may consider the polynomial ring $K[x_1, x_2, ..., x_n]$ over the $n$ variables $x_1, x_2, ..., x_n$ where the elements of $K[x_1, x_2, ..., x_n]$ are polynomials in the indeterminants $x_1, x_2, ..., x_n$ with coefficients in $K$. For example, the following polynomial $F$ is contained in $\mathbb{F}_3[x_1, x_2]$:

(1)

\begin{align} \quad F(x_1, x_2) = 2x_1 + x_2 \end{align}

Definition: Let $K$ be a field and let $F \in K[x_1, x_2, ..., x_n]$. A point $\mathbf{p} = (p_1, p_2, ..., p_n) \in \mathbb{A}^n$ is said to be a Zero or Root of $F$ if $F(\mathbf{p}) = 0$.

In the example above we have that $\mathbf{p} = (1, 2) \in \mathbb{A}^2(\mathbb{F}_3)$ is a root of $F$ since:

(2)

\begin{align} \quad F(1, 2) = 2(1) + (2) = 3 \equiv 0 \pmod 3 \end{align}

Of course, roots need not be unique. We have that the point $\mathbf{q} = (0, 0) \in \mathbb{A}^2(\mathbb{F}_3)$ is also a root of $F$ since:

(3)

\begin{align} \quad F(0, 0) = 2(0) + (0) = 0 \end{align}