Introduction to algebraic geometry

Let $\mathbb{K}$ be a fixed body (in this article, the $\mathbb{Q}$ body of rational numbers, the $ℝ$ body of real numbers or the $ℂ$ body of complex numbers).

Consider a system of equations P_i(X₁,..., X_n) = 0 where the P_i are polynomials with coefficients in $\mathbb{K}$ and the variables take their values in $\mathbb{K}$ (or a body containing $\mathbb{K}$). Typically, the equations will have integer coefficients, and we will consider the solutions of the system in ${ℝ}^n$ or ${ℂ}^n\mathrm{.}$

Such a system defines a subset $\mathrm{X}\subset {\mathbb{K}}^n$ called an algebraic variety, and algebraic geometry consists in understanding the "geometric" properties of X using the "algebraic" properties of the system of equations. A fundamental historical example is the study of the trajectories of planets and comets, which are, to a first approximation, plane curves of the second degree (i.e. conics: ellipses, parabolas or hyperbolas).

Although in practice we're interested (as in the example above) in the real case, we find that the relationship sought between algebra and geometry is much more satisfactory in the case of the body of complex numbers (fundamentally because a one-variable polynomial of degree n with real or complex coefficients has exactly n "distinct or confused" complex roots, whereas if it has real coefficients (e.g. integers), the number of its real roots (distinct or confused) can be any number between...