In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
. This relationship is the basis of
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. It relates
algebraic sets to
ideals in
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
s over
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
s. This relationship was discovered by
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
, who proved the Nullstellensatz in his second major paper on
invariant theory in 1893 (following his seminal 1890 paper in which he proved
Hilbert's basis theorem
In mathematics Hilbert's basis theorem asserts that every ideal (ring theory), ideal of a polynomial ring over a field (mathematics), field has a finite generating set of an ideal, generating set (a finite ''basis'' in Hilbert's terminology).
In ...
).
Formulation
Let
be a
field (such as the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s) and
be an algebraically closed
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
of
(such as the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s). Consider the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...