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## Algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations.
## Zeroes of simultaneous polynomials
In algebraic geometry, the geometric objects studied are defined as the set of zeroes of a number of polynomials: meaning the set of common zeroes, or equally the set defined by one or several simultaneous polynomial equations. For instance, the two-dimensional sphere in three-dimensional Euclidean space *x*^{2}+*y*^{2}+*z*^{2}-1 = 0.
R^{3} can be defined as the set of all points (x, y, z) which satisfy the two polynomial equations
*x*^{2}+*y*^{2}+*z*^{2}-1 = 0*x*+*y*+*z*= 0
## Affine varieties
In general, if ## Coordinate ring of a variety
To every variety ## Projective theory
Instead of working in the affine space ## Background to the current point of view on the subjectIn the modern view, the correspondence between variety and coordinate ring is turned around: one starts with an abstract commutative ring and defines a corresponding variety via its prime ideals. The prime ideals are first turned into a topological space, the spectrum of the ring. In the most general formulation, this leads to Alexander Grothendieck's schemess. An important class of varieties are the abelian varieties which are varieties whose points form an abelian group. The prototypical examples are the elliptic curves that were instrumental in the proof of Fermat's last theorem and are also used in elliptic curve cryptography. While much of algebraic geometry is concerned with abstract and general statements about varieties, methods for the effective computation with concretely given polynomials have also been developed. The most important is the technique of Gröbner bases which is employed in all computer algebra systems. Algebraic geometry was developed largely by the Italian geometers in the early part of the 20th century. Their work on birational geometry was deep; but didn't rest on a sufficiently rigorous basis. Commutative algebra (as the study of commutative rings and their ideals) was developed by David Hilbert, Emmy Noether and others, also in the 20h century, with the geometric applications in mind. In the 1930s and 1940s Oscar Zariski, André Weil and others realized that a requirement existed for an axiomatic algebraic geometry on a rigorous basis. For a while there were several foundational theories used. In the 1950s and 1960s Jean-Pierre Serre and Grothendieck recast the foundations making use of the theory of sheaves. Later, from about 1960, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilised in the 1970s, and applications were made, both to number theory and to more classical geometric questions on algebraic varieties, singularities and moduli.
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