|
Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem" – see ''Satz'') is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert who proved the Nullstellensatz and several other important related theorems named after him (like Hilbert's basis theorem). == Formulation == Let ''k'' be a field (such as the rational numbers) and ''K'' be an algebraically closed field extension (such as the complex numbers), consider the polynomial ring ''k''(''X''''n'' ) and let ''I'' be an ideal in this ring. The algebraic set V(''I'') defined by this ideal consists of all ''n''-tuples x = (''x''1,...,''x''''n'') in ''K''''n'' such that ''f''(x) = 0 for all ''f'' in ''I''. Hilbert's Nullstellensatz states that if ''p'' is some polynomial in ''k''(''X''''n'' ) that vanishes on the algebraic set V(''I''), i.e. ''p''(x) = 0 for all x in ''V''(''I''), then there exists a natural number ''r'' such that ''p''''r'' is in ''I''. An immediate corollary is the "weak Nullstellensatz": The ideal ''I'' in ''k''(''X''''n'' ) contains 1 if and only if the polynomials in ''I'' do not have any common zeros in ''K''''n''. It may also be formulated as follows: if ''I'' is a proper ideal in ''k''(''X''''n'' ), then V(''I'') cannot be empty, i.e. there exists a common zero for all the polynomials in the ideal in every algebraically closed extension of ''k''. This is the reason for the name of the theorem, which can be proved easily from the 'weak' form using the Rabinowitsch trick. The assumption of considering common zeros in an algebraically closed field is essential here; for example, the elements of the proper ideal (''X''2 + 1) in R() do not have a common zero in R. With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as : for every ideal ''J''. Here, denotes the radical of ''J'' and I(''U'') is the ideal of all polynomials that vanish on the set ''U''. In this way, we obtain an order-reversing bijective correspondence between the algebraic sets in ''K''''n'' and the radical ideals of ''K''(''X''''n'' ). In fact, more generally, one has a Galois connection between subsets of the space and subsets of the algebra, where "Zariski closure" and "radical of the ideal generated" are the closure operators. As a particular example, consider a point . Then . More generally, : Conversely, every maximal ideal of the a polynomial ring (note that is algebraically closed) is of the form for some . As another example, an algebraic subset ''W'' in ''K''''n'' is irreducible (in the Zariski topology) if and only if is a prime ideal. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hilbert's Nullstellensatz」の詳細全文を読む スポンサード リンク
|