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One of the main objects of mathematics is to solve, explicitly, systems of equations. In the first place are algebraic systems. Diophantian equations are among the most difficult. Since solutions are hard to compute (think of Fermat's Theorem), one approach is to study solutions as a whole set, let us say, as a geometrical object. But what kind of geometry to deal with in case of diophantian equations over integer numbers (instead of the usual real or complex ones)? The answer is given by the theory of schemes in algebraic geometry. In its end state, it is due to Alexandre Grothendieck, whose first goal was to solve Weil's conjectures.

The main idea in schemes theory is to see any commutative ring with $1$ (e.g. $\mathbb Z$ ) as the set of "algebraic functions" on some geometrical object (to be defined, called spectrum of the ring).

There are many good books on the subject:

1) The Red Book of Varieties and Schemes, by D. Mumford,

2) Basic Algebraic Geometry, by I.R.Shafarevich,

3) Algebraic Geometry, by R. Hartshorne,

4) The Geometry of Schemes, by D.Eisenbud and J.Harris.

To learn schemes, one has to know first some commutative algebra. The book Introduction to Commutative Algebra by M.F. Atiyah and I.G.MacDonald is excellent (to begin with). There are also, by H. Matsumura, Commutative Algebra, and by D. Eisenbud, Commutative Algebra (With a view toward Algebraic Geometry).

The foundations of the theory by A. Grothendieck are developed in the EGA's (Eléments de Géométrie Algébrique, by Grothendieck & Dieudonné, Publ.Math. IHES; EGA (I) is a Springer-book), and in the SGA's (Séminaire de Géométrie Algébrique, by Grothendieck & al., Springer or SMF).

Answers in mathoverflow.net/questions/59071/what-elementary-problems-can-you-solve-with-schemes give many simple and significant applications of schemes.

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