rational points of an algebraic variety

In http://en.wikipedia.org/wiki/Rational_point we read :

a $K$-rational point is a point on an algebraic variety where each coordinate of the point >belongs to the field $K$. This means that, if the variety is given by a set of equations

$$f_i(x_1, ..., x_n)=0,\; j=1, \cdots, m$$

then the $K$-rational points are solutions $(x_1, ..., x_n)\in K^n$ of the equations

My understanding is that the algebraic variety is the set of solutions $(x_1, ..., x_n)\in K^n$ of the equations so by definition all points of the algebraic variety are $K$-rational points.. I'm I wrong? so what is the meaning of defining rational points if all points of ther alg variety are rational?

The notion of a $K$-rational point depends very much on the field $K$. For example, consider the algebraic variety given as the zero set of the polynomial $x^2+y^2+1\in \mathbb{C}[x,y]$. Note that this variety has no $\mathbb{R}$-rational points.

Exercise 1: Prove that if we consider the algebraic variety given as the zero set of the polynomial $x^2+y^2-1\in \mathbb{R}[x,y]$, then this variety has infinitely many $\mathbb{Q}$-rational points.

In light of Exercise 1, note that the set of $\mathbb{Q}$-rational points of this variety is a proper subset of the set of $\mathbb{R}$-rational points of this variety.

In general, if $X$ is a scheme over a field $K$, then we can speak of the set of all $K$-rational points of $X$. Note that the $K$-rational points of $X$ are precisely those points of $X$ at which the residue field of the local ring at that point is isomorphic to $K$.

Definition: Let $\pi:X\to S$ be an $S$-scheme. A $\textit{section}$ of $X$ is a morphism of $S$-schemes $\sigma:S\to X$. This amounts to saying that $\pi\circ\sigma=\text{Id}_S$. The set of sections of $X$ is denoted by $X(S)$ (and also by $X(A)$ if $S=\text{Spec}(A))$).

Exercise 2: Prove that if $X$ is a scheme over a field $K$, then we can identify $X(K)$ with the set of all $K$-rational points of the scheme $X$.

Let $K$ be a field and let $X=\text{Spec }K[T_1,\dots,T_n]/I$ be an affine scheme over $K$. Let $Z$ be the zero set of some polynomial $P\in I$. It is natural to ask whether our notion of a "$K$-rational point of $X$" in the scheme-theoretic sense agrees with our naive notion of a $K$-rational point of the algebraic set defined by $P$.

Exercise 3: Prove that we have a canonical bijection $Z\to X(K)$ where $X(K)$ denotes the set of all $K$-rational points of the scheme $X$.

An algebraic variety (over a non-algebraically-closed field) is "more than just a set of points". If X is an algebraic variety over $K$, you can make sense of the set $X(L)$ of $L$-points for any field extension $L / K$.

For instance, consider the empty subvariety of $\mathbb{Q}^1$ (defined, if you like, by the equation $1 = 0$) and the subvariety defined by $X^2 + 1 = 0$. Both of these have no $\mathbb{Q}$-points, but their $\mathbb{C}$-points are very different!

If you want to think of a variety as a set of points, it's best to think of it as a set of points in $\overline{K}^n$, where $\overline{K}$ is the algebraic closure of $K$. Then you can meaningfully ask which ones of these are contained in $K^n$.

For example Hartshorne defines affine variety as the set of solutions in $K^n$ only in the case that $K$ is algebraically closed. I rather think that this is the norm. Without this assumption a lot of the basic stuff goes out of the window. Nullstellensatz for example. As an extreme example would you like to treat the origin, a single point, as the solution set of the equation $x^2+y^2=0$ over the reals, when we `know' that the set of (affine) solutions really is the union of the two lines $x=iy$ and $x=-iy$.

You can study geometric objects associated with polynomial equations (with coefficients in $K$) over a non-algebraically closed field $K$, but then you need to use the language of schemes or a suitable substitute. Then we get "non-rational points" or points of degree $>1$. This way of bookkeeping (keeping track of the degree of the field extension $K(x_1,\ldots,x_n)/K$, where the coordinates of a solution are $(x_1,\ldots,x_n)$ ) gives us uniform answers to several questions. For example, consider the curve $x^2+y^2=1$ over the field of rational numbers. Superficially it would look like there are 2 points on that curve such that $x=3/5$, namely $(x,y)=(3/5,4/5)$ and $(3/5,-4/5)$, but no points with $x$-coordinate equal to $1/2$. With this bookkeeping in place we would say that there is a single degree two point with $x=1/2$, namely that with $y=\pm\sqrt{3}/2$. Even though this may look like two points also, it is treated as a single degree two point. This is because we cannot distinguish between the two points in the sense that if $P(u,v)$ is polynomial with rational coefficents, then $P(1/2,\sqrt3/2)$ and $P(1/2,-\sqrt3/2)$ are both zero or neither is. Branch points (=ramification) cause another change to the way of counting, but in the end we get `two points' with a given $x$-coordinate with appropriate bookkeeping.

Basically the point is that some fields are too small to tell us the whole story of the set of solutions of a system of polynomial equations. I may not have listed the most natural reasons. Expect many answers to this one.