User:Maximilian Janisch/latexlist/Algebraic Groups/Hilbert theorem

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Hilbert's basis theorem

If $A$ is a commutative Noetherian ring and $A[X_1,\ldots,X_n]$ is the ring of polynomials in $X_1,\ldots,X_n$ with coefficients in $A$, then $A[X_1,\ldots,X_n]$ is also a Noetherian ring. In particular, in a ring of polynomials in a finite number of variables over a field or over a ring of integers any ideal is generated by a finite number of elements (has a finite basis). This is the form in which the theorem was demonstrated by D. Hilbert [1]; it was used as auxiliary theorem in the proof of Hilbert's theorem on invariants (see below, 8). Subsequently, Hilbert's basis theorem was extensively used in commutative algebra.

References

[1]	D. Hilbert, "Ueber die Theorie der algebraischen Formen" Math. Ann. , 36 (1890) pp. 473–534 MR1510634 Zbl 22.0133.01

V.I. Danilov

Hilbert's irreducibility theorem

Let $f ( t _ { 1 } , \ldots , t _ { k } , x _ { 1 } , \ldots , x _ { N } )$ be an irreducible polynomial over the field $0$ of rational numbers; then there exists an infinite set of values $t _ { 1 } ^ { 0 } , \ldots , t _ { x } ^ { 0 } \in Q$ of the variables $t _ { 1 } , \ldots , t _ { k }$ for which the polynomial $f ( t _ { 1 } ^ { 0 } , \ldots , t _ { x } ^ { 0 } , x _ { 1 } , \ldots , x _ { x } )$ is irreducible over $0$. Thus, the polynomial $f ( t , x ) = t - x ^ { 2 }$ remains irreducible for all $t ^ { 0 }$ ($t ^ { 0 } \neq a ^ { 2 }$, $a \in 0$) and only for them. This theorem, which was obtained by D. Hilbert in 1892, was subsequently generalized to the case of polynomials over certain other fields (e.g., over a field of finite type over its prime subfield [2]).

Hilbert's irreducibility theorem is employed in investigations connected with the inverse problem in Galois theory and with the arithmetic of algebraic varieties (cf. Galois theory, inverse problem of; Algebraic varieties, arithmetic of). Let there exist an extension $E / K$ with Galois group $k$ over the field $K = k ( t _ { 1 } , \ldots , t _ { n } )$ of rational functions in $t _ { 1 } , \ldots , t _ { x }$, with $k$ an algebraically closed field in $k$ such that Hilbert's irreducibility theorem is applicable to it. Then it is possible to choose values of the variables $t _ { 1 } , \ldots , t _ { x }$ in $k$ such that the obtained extension of $k$ has Galois group $k$. With the aid of this concept Hilbert constructed [1] extensions of $0$ with a symmetric and an alternating group; in the case of the symmetric group $k$ is taken to be the field of rational functions in $12$ variables, while $K$ is a subfield of the field of symmetric functions, which is itself a field of rational functions. In a generalization of this approach, E. Noether considered an arbitrary subgroup $G \subset S _ { Y }$ and the extension of $k$ of the corresponding field of invariants of $k$ with respect to $k$ [3]. Hilbert's irreducibility theorem makes it possible to construct an extension of $k$ with Galois group $k$, as long as $E ^ { G }$ is a field of rational functions over $0$. The problem of satisfying this condition (Noether's problem) is closely connected with the Lüroth problem. Only in 1969 was it shown by R. Swan that the answer to the problem is negative in most cases [4], [6].

Hilbert's irreducibility theorem is also employed in constructing rational points of Abelian varieties $4$ over the field $0$ of rational numbers. By the Mordell–Weil theorem, the group of rational points of $4$ is finitely generated and there arises the question of the value of its rank $N$. Using Hilbert's irreducibility theorem, A. Neron constructed varieties $4$ of dimension and rank higher than or equal to $3 g + 6$ [2].

References

[1]	D. Hilbert, "Ueber die Irreducibilität ganzer rationaler Funktionen mit ganzzahligen Koefficienten" J. Reine Angew. Math. , 110 (1892) pp. 104–129
[2]	S. Lang, "Diophantine geometry" , Interscience (1962) MR0142550 Zbl 0115.38701
[3]	N.G. Chebotarev, "Galois theory" , Moscow-Leningrad (1936) pp. 18–32 (In Russian)
[4]	J. Martinet, "Un contre-exemple à une conjecture d'E. Noether (d'apres R. Swan)" , Sem. Bourbaki , 22 : 372 (1969–1970)
[5]	A. Schinzel, "Reducibility of polynomials" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 491–496 MR0424768 Zbl 0233.12101 Zbl 0228.12101 Zbl 0223.12103 Zbl 0219.12051
[6]	V.E. Voskresenskii, "The geometry of linear algebraic groups" Proc. Steklov Inst. Math. , 132 (1975) pp. 178–183 Trudy Mat. Inst. Steklov. , 132 (1973) pp. 151–161 MR0342521 Zbl 0309.14040

A.N. Parshin

Hilbert's Nullstellen Satz

Hilbert's zero theorem, Hilbert's root theorem

Let $k$ be a field, let $k [ X _ { 1 } , \ldots , X _ { x } ]$ be a ring of polynomials over $k$, let $k$ be the algebraic closure of $k$, and let $F , F _ { 1 } , \ldots , F _ { m }$ be polynomials in $k [ X _ { 1 } , \ldots , X _ { x } ]$. A root of the polynomial $F ( X _ { 1 } , \dots , X _ { p } )$ is a sequence $( c _ { 1 } , \dots , c _ { r } )$ of elements in $k$ satisfying the condition $F ( c _ { 1 } , \dots , c _ { m } ) = 0$. If each common root of the polynomials $F _ { 1 } , \ldots , F _ { m }$ is a root of the polynomial $H ^ { \prime }$, then there exists an integer $N$, depending only on $F _ { 1 } , \ldots , F _ { m }$, such that $F ^ { \prime }$ belongs to the ideal generated by $F _ { 1 } , \ldots , F _ { m }$, i.e.

\begin{equation} F ^ { \gamma } = A _ { 1 } F _ { 1 } + \ldots + A _ { m } F _ { m } \end{equation}

where $A _ { 1 } , \dots , A _ { m }$ are certain polynomials. This result has been obtained by D. Hilbert [1].

The theorem is equivalent to the statement that for any proper ideal $1$ of the ring $k [ X _ { 1 } , \ldots , X _ { x } ]$ there exists a root which is common to all polynomials in $1$. Thus, this theorem may be regarded as a far-reaching generalization of the fundamental theorem of algebra (cf. Algebra, fundamental theorem of). It may also be regarded as the statement that any prime ideal of the ring $k [ X _ { 1 } , \ldots , X _ { x } ]$ is the intersection of the maximal ideals which contain it; this leads to the concept of a Jacobson ring.

In the geometric interpretation, the roots of an ideal $\mathfrak { a } \subset k [ X _ { 1 } , \ldots , X _ { n } ]$ correspond to the algebraic points of the affine variety defined by $1$. Hilbert's theorem implies that there exists an algebraic point in any non-empty affine variety. Thus, the set of algebraic points is everywhere dense on the variety and thus uniquely defines it — which is the reason why one often restricts oneself to algebraic points when studying algebraic varieties.

References

[1]	D. Hilbert, "Ueber die vollen Invariantensysteme" Math. Ann. , 42 (1893) pp. 313–373 MR1510781 Zbl 25.0173.01
[2]	B.L. van der Waerden, "Algebra" , 2 , Springer (1971) (Translated from German) MR1541390 Zbl 0221.12001
[3]	O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001
[4]	S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[5]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) MR0360549 Zbl 0279.13001

V.I. Danilov

Hilbert's theorem on surfaces of negative curvature

In the three-dimensional Euclidean space there is no complete regular surface of constant negative curvature. Demonstrated by D. Hilbert [1] in 1901.

References

[1]	D. Hilbert, "Grundlagen der Geometrie" , Springer (1913) MR1859422 MR1807508 MR1807507 MR1732507 MR1676305 MR1109913 MR0981143 MR0874532 MR0851072 MR0799771 MR0474006 MR0309913 MR0262046 MR0229120 MR0177322 MR0098003 MR0080308 MR1511181 Zbl 44.0543.02

E.V. Shikin

Hilbert's syzygies theorem

A theorem on finiteness of a chain of syzygies) of a graded module over a ring of polynomials (for the classical formulation see [1]).

Let $A$ be a Noetherian ring, let $M$ be a Noetherian $A$-module and let $x_1,\ldots,x_n$ be a system of generators of $M$. The module of syzygies (relations) $S(M)$ of $M$ is the module of relations for $x_1,\ldots,x_n$, i.e. the $A$-module of vectors $(a_1,\ldots,a_n)$, $a_i \in A$, which satisfy the condition $a_1x_1 + \cdots + a_n x_n = 0$. Inductively one defines the $i$-th module of syzygies by $S_i(M) = S(S_{i-1}(M))$ (where $S_0(M) = M$). It may also be described in a different manner as an exact sequence, known as a chain of syzygies: $$ 0 \rightarrow S_i(M) \rightarrow F_{i-1 } \rightarrow \cdots \rightarrow F_0 \rightarrow M \rightarrow 0 $$ where $F_0,\ldots,F_{i-1}$ are free $A$-modules of finite type.

In its modern interpretation, Hilbert's syzygies theorem is formulated as follows: If $A$ is a local regular ring of dimension $m$, then the $m$-th module of syzygies of an arbitrary Noetherian $A$-module is a free module. This is equivalent to saying that any $A$-module has a free resolution of length $m$ or that $A$ has global projective dimension $m$. This property is characteristic of regular rings [2].

The global variant of Hilbert's syzygies theorem: Over a regular ring $A$ (e.g. over a ring of polynomials) any $A$-module of finite type has a projective resolution (but not necessarily free) of finite length.

References

[1]	D. Hilbert, "Ueber die Theorie der algebraischen Formen" Math. Ann. , 36 (1890) pp. 473–534 MR1510634 Zbl 22.0133.01
[2]	J.-P. Serre, "Sur la dimension homologique des anneaux et des modules noethériens" S. Iyanaga (ed.) Y. Kawada (ed.) , Proc. Internat. Symp. Algebraic Number Theory , Sci. Council Tokyo (1955) pp. 175–189 MR0086071 Zbl 0073.26004
[3]	J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965) MR0201468 Zbl 0142.28603
[4]	O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001

V.I. Danilov

Hilbert's theorem on cyclic extensions

Hilbert's theorem $90$)

Let $K$ be a cyclic extension of a field $k$ with cyclic Galois group $G ( K / k )$ and let $\Omega$ be the generator of $G ( K / k )$; the norm $N _ { K / k } ( \beta )$ of an element $\beta \in K$ is then equal to one if and only if there exists a non-zero element $\alpha \in K$ satisfying the condition $\beta = \alpha \cdot \sigma ( \alpha ) ^ { - 1 }$. In a similar manner, the trace $\operatorname { Tr } _ { K / k } ( \beta )$ is zero if and only if $3$ can be represented in the form $\beta = \alpha - \sigma ( \alpha )$, $\alpha \in K$, [1], [2], [3].

Hilbert's theorem may be considered as a consequence of a more general theorem on the cohomology of Galois groups [4]. In fact, if $K$ is a Galois extension of a field $k$ with Galois group $k$, then the multiplicative group $K ^ { * }$ of $K$ has the structure of a $k$-module, and the first cohomology group $H ^ { 1 } ( G , K ^ { * } )$ vanishes. In the same manner, if $q \geq 1$, $H ^ { q } ( G , K ) = 0$ (cf. Galois cohomology).

Another generalization of Hilbert's theorem is Grothendieck's descent theorem; one of its applications in étale topology, which is also known as Hilbert's theorem $90$, states that the étale cohomology groups $H ^ { 1 } ( X _ { et } , G _ { m } )$ of a scheme $x$ with values in a sheaf of multiplicative groups $G _ { m }$ is isomorphic to the Picard group $\operatorname { Pic } ( X )$ of classes of invertible sheafs on $x$ [5].

References

[1]	D. Hilbert, "Die Theorie der algebraischen Zahlkörper" Jahresber. Deutsch. Math.-Verein , 4 (1897) pp. 175–546 Zbl 28.0157.05
[2]	S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[3]	N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207
[4]	J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0180551 Zbl 0128.26303
[5]	M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , Théorie des topos et cohomologie étale des schémas (SGA 4) , Lect. notes in math. , 269; 270; 305 , Springer (1972–1973)

V.I. Danilov

Hilbert's theorem on the existence of an absolute extremum

Let

\begin{equation} I = \int F d t \end{equation}

be the functional of a variational problem in parametric form, where $F = F ( x , y , \dot { x } , \dot { y } )$ is a positive-definite function of the first degree in $( \dot { x } , \dot { y } )$ which is three times continuously differentiable with respect to all arguments for all $( x , y )$ from a domain $k$ and all $( \dot { x } , \dot { y } )$ which meet the condition $\dot { x } \square ^ { 2 } + \dot { y } \square ^ { 2 } \neq 0$. It also assumed that $F ( x , y , \xi , \eta ) > 0$ for all $( x , y ) \in G$ and all $( \xi , \eta )$ with $\xi ^ { 2 } + \eta ^ { 2 } = 1$ (i.e. the functional $1$ is positive definite), and also that the sets $( x , y ) = \{ ( \xi , \eta ) : F ( x , y , \xi , \eta ) \leq 1 \}$ are strictly convex with respect to $( \xi , \eta )$ for all $( x , y )$ in a closed convex subdomain $G$ (i.e. the functional $1$ is regular or elliptic).

Under the above assumptions it is possible to find for any two points $( x _ { 0 } , y _ { 0 } )$ and $( x _ { 1 } , y _ { 1 } )$ in $G$ a curve which is the absolute minimum over all rectifiable curves for $1$.

The theorem was obtained by D. Hilbert in 1899.

References

[1]	N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian) MR0142019 Zbl 0718.49001

V.M. Tikhomirov

Hilbert's theorem on invariants

A theorem that establishes that the algebra of all polynomials on the complex vector space of forms of degree $a$ in $N$ variables which are invariant with respect to the action of the general linear group $GL ( r , C )$, defined by linear substitutions of these variables, is finitely generated. The first proof of the theorem using Hilbert's basis theorem as well as formal processes of the theory of invariants, was given in [1] (cf. also Invariants, theory of). D. Hilbert [2] gave a constructive proof of this theorem.

Hilbert's theorem is the first fundamental theorem of the theory of invariants for the $a$-th symmetric degree of the standard representation of $GL ( r , C )$. The proof of Hilbert's theorem stimulated the formulation of the problem of finite generation of algebras of invariants for subgroups of $GL ( r , C )$ and also the formulation of Hilbert's 14th problem. It was proved by H. Weyl, who employed the theory of integration on groups, that the algebra of invariants is finitely generated for any finite-dimensional representation of a compact Lie group or a complex semi-simple Lie group [3].

Hilbert's theorem is also the name usually given to the following generalization. If $R$ is an algebra of finite type over a field $k$, if $k$ is the geometrically reductive group of its $k$-automorphisms and if $R ^ { G }$ is the subalgebra of all $k$-invariant elements in $R$, then $R ^ { G }$ is also of finite type over $k$ [4], [5].

References

[1]	D. Hilbert, "Ueber die Theorie der algebraischen Formen" Math. Ann. , 36 (1890) pp. 473–534 MR1510634 Zbl 22.0133.01
[2]	D. Hilbert, "Ueber die vollen Invariantensysteme" Math. Ann. , 42 (1893) pp. 313–373 MR1510781 Zbl 25.0173.01
[3]	H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) MR0000255 Zbl 1024.20502
[4]	D. Mumford, "Geometric invariant theory" , Springer (1965) MR0214602 Zbl 0147.39304
[5]	M. Nagata, "Invariants of a group in an affine ring" J. Math. Kyoto Univ. , 3 (1964) pp. 369–377 MR0179268 Zbl 0146.04501