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==Hilbert's basis theorem==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h0474101.png" /> is a commutative [[Noetherian ring|Noetherian ring]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h0474102.png" /> is the ring of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h0474103.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h0474104.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h0474105.png" /> 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 [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Hilbert, "Ueber die Theorie der algebraischen Formen" ''Math. Ann.'' , '''36''' (1890) pp. 473–534 {{MR|1510634}} {{ZBL|22.0133.01}} </TD></TR></table>

''V.I. Danilov''

==Hilbert's irreducibility theorem==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h0474106.png" /> be an irreducible polynomial over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h0474107.png" /> of rational numbers; then there exists an infinite set of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h0474108.png" /> of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h0474109.png" /> for which the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741010.png" /> is irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741011.png" />. Thus, the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741012.png" /> remains irreducible for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741015.png" />) 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 [[#References|[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|Galois theory, inverse problem of]]; [[Algebraic varieties, arithmetic of|Algebraic varieties, arithmetic of]]). Let there exist an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741016.png" /> with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741017.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741018.png" /> of rational functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741019.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741020.png" /> an algebraically closed field in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741021.png" /> such that Hilbert's irreducibility theorem is applicable to it. Then it is possible to choose values of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741023.png" /> such that the obtained extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741024.png" /> has Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741025.png" />. With the aid of this concept Hilbert constructed [[#References|[1]]] extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741026.png" /> with a symmetric and an alternating group; in the case of the symmetric group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741027.png" /> is taken to be the field of rational functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741028.png" /> variables, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741029.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741030.png" /> and the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741031.png" /> of the corresponding field of invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741032.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741033.png" /> [[#References|[3]]]. Hilbert's irreducibility theorem makes it possible to construct an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741034.png" /> with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741035.png" />, as long as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741036.png" /> is a field of rational functions over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741037.png" />. The problem of satisfying this condition (Noether's problem) is closely connected with the [[Lüroth problem|Lüroth problem]]. Only in 1969 was it shown by R. Swan that the answer to the problem is negative in most cases [[#References|[4]]], [[#References|[6]]].

Hilbert's irreducibility theorem is also employed in constructing rational points of Abelian varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741038.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741039.png" /> of rational numbers. By the Mordell–Weil theorem, the group of rational points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741040.png" /> is finitely generated and there arises the question of the value of its rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741041.png" />. Using Hilbert's irreducibility theorem, A. Neron constructed varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741042.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741043.png" /> and rank higher than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741044.png" /> [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Hilbert, "Ueber die Irreducibilität ganzer rationaler Funktionen mit ganzzahligen Koefficienten" ''J. Reine Angew. Math.'' , '''110''' (1892) pp. 104–129 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang, "Diophantine geometry" , Interscience (1962) {{MR|0142550}} {{ZBL|0115.38701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N.G. Chebotarev, "Galois theory" , Moscow-Leningrad (1936) pp. 18–32 (In Russian) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J. Martinet, "Un contre-exemple à une conjecture d'E. Noether (d'apres R. Swan)" , ''Sem. Bourbaki'' , '''22''' : 372 (1969–1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A. Schinzel, "Reducibility of polynomials" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars (1971) pp. 491–496 {{MR|0424768}} {{ZBL|0233.12101}} {{ZBL|0228.12101}} {{ZBL|0223.12103}} {{ZBL|0219.12051}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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 {{MR|0342521}} {{ZBL|0309.14040}} </TD></TR></table>

''A.N. Parshin''

==Hilbert's Nullstellen Satz==
''Hilbert's zero theorem, Hilbert's root theorem''

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741045.png" /> be a field, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741046.png" /> be a ring of polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741047.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741048.png" /> be the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741049.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741050.png" /> be polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741051.png" />. A root of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741052.png" /> is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741053.png" /> of elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741054.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741055.png" />. If each common root of the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741056.png" /> is a root of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741057.png" />, then there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741058.png" />, depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741059.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741060.png" /> belongs to the ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741061.png" />, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741062.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741063.png" /> are certain polynomials. This result has been obtained by D. Hilbert [[#References|[1]]].

The theorem is equivalent to the statement that for any proper ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741064.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741065.png" /> there exists a root which is common to all polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741066.png" />. Thus, this theorem may be regarded as a far-reaching generalization of the fundamental theorem of algebra (cf. [[Algebra, fundamental theorem of|Algebra, fundamental theorem of]]). It may also be regarded as the statement that any prime ideal of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741067.png" /> is the intersection of the maximal ideals which contain it; this leads to the concept of a [[Jacobson ring|Jacobson ring]].

In the geometric interpretation, the roots of an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741068.png" /> correspond to the algebraic points of the affine variety defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h04741069.png" />. 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Hilbert, "Ueber die vollen Invariantensysteme" ''Math. Ann.'' , '''42''' (1893) pp. 313–373 {{MR|1510781}} {{ZBL|25.0173.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.L. van der Waerden, "Algebra" , '''2''' , Springer (1971) (Translated from German) {{MR|1541390}} {{ZBL|0221.12001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR|0360549}} {{ZBL|0279.13001}} </TD></TR></table>

''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 [[#References|[1]]] in 1901.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Hilbert, "Grundlagen der Geometrie" , Springer (1913) {{MR|1859422}} {{MR|1807508}} {{MR|1807507}} {{MR|1732507}} {{MR|1676305}} {{MR|1109913}} {{MR|0981143}} {{MR|0874532}} {{MR|0851072}} {{MR|0799771}} {{MR|0474006}} {{MR|0309913}} {{MR|0262046}} {{MR|0229120}} {{MR|0177322}} {{MR|0098003}} {{MR|0080308}} {{MR|1511181}} {{ZBL|44.0543.02}} </TD></TR></table>

''E.V. Shikin''

==Hilbert's syzygies theorem==
A theorem on finiteness of a chain of [[Syzygy|syzygies]]) of a graded module over a ring of polynomials (for the classical formulation see [[#References|[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 module|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 [[#References|[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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> D. Hilbert, "Ueber die Theorie der algebraischen Formen" ''Math. Ann.'' , '''36''' (1890) pp. 473–534 {{MR|1510634}} {{ZBL|22.0133.01}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> 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 {{MR|0086071}} {{ZBL|0073.26004}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> J.-P. Serre, "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer (1965) {{MR|0201468}} {{ZBL|0142.28603}} </TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </TD></TR>
</table>

''V.I. Danilov''

==Hilbert's theorem on cyclic extensions==
''Hilbert's theorem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410100.png" />)''

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410101.png" /> be a cyclic extension of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410102.png" /> with cyclic Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410103.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410104.png" /> be the generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410105.png" />; the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410106.png" /> of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410107.png" /> is then equal to one if and only if there exists a non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410108.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410109.png" />. In a similar manner, the trace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410110.png" /> is zero if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410111.png" /> can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410112.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410113.png" />, [[#References|[1]]], [[#References|[2]]], [[#References|[3]]].

Hilbert's theorem may be considered as a consequence of a more general theorem on the cohomology of Galois groups [[#References|[4]]]. In fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410114.png" /> is a [[Galois extension|Galois extension]] of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410115.png" /> with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410116.png" />, then the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410117.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410118.png" /> has the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410119.png" />-module, and the first cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410120.png" /> vanishes. In the same manner, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410122.png" /> (cf. [[Galois cohomology|Galois cohomology]]).

Another generalization of Hilbert's theorem is Grothendieck's descent theorem; one of its applications in [[Etale topology|étale topology]], which is also known as Hilbert's theorem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410124.png" />, states that the étale cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410125.png" /> of a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410126.png" /> with values in a sheaf of multiplicative groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410127.png" /> is isomorphic to the Picard group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410128.png" /> of classes of invertible sheafs on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410129.png" /> [[#References|[5]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Hilbert, "Die Theorie der algebraischen Zahlkörper" ''Jahresber. Deutsch. Math.-Verein'' , '''4''' (1897) pp. 175–546 {{MR|}} {{ZBL|28.0157.05}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) {{MR|0354207}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0180551}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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)</TD></TR></table>

''V.I. Danilov''

==Hilbert's theorem on the existence of an absolute extremum==

Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410130.png" /></td> </tr></table>

be the functional of a variational problem in parametric form, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410131.png" /> is a positive-definite function of the first degree in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410132.png" /> which is three times continuously differentiable with respect to all arguments for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410133.png" /> from a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410134.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410135.png" /> which meet the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410136.png" />. It also assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410137.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410138.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410139.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410140.png" /> (i.e. the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410141.png" /> is positive definite), and also that the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410142.png" /> are strictly convex with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410143.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410144.png" /> in a closed convex subdomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410145.png" /> (i.e. the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410146.png" /> is regular or elliptic).

Under the above assumptions it is possible to find for any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410147.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410148.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410149.png" /> a curve which is the absolute minimum over all rectifiable curves for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410150.png" />.

The theorem was obtained by D. Hilbert in 1899.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian) {{MR|0142019}} {{ZBL|0718.49001}} </TD></TR></table>

''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410151.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410152.png" /> variables which are invariant with respect to the action of the general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410153.png" />, 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 [[#References|[1]]] (cf. also [[Invariants, theory of|Invariants, theory of]]). D. Hilbert [[#References|[2]]] gave a constructive proof of this theorem.

Hilbert's theorem is the first fundamental theorem of the theory of invariants for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410154.png" />-th symmetric degree of the standard representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410155.png" />. The proof of Hilbert's theorem stimulated the formulation of the problem of finite generation of algebras of invariants for subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410156.png" /> 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 [[#References|[3]]].

Hilbert's theorem is also the name usually given to the following generalization. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410157.png" /> is an algebra of finite type over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410158.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410159.png" /> is the geometrically [[Reductive group|reductive group]] of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410160.png" />-automorphisms and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410161.png" /> is the subalgebra of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410162.png" />-invariant elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410163.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410164.png" /> is also of finite type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047410/h047410165.png" /> [[#References|[4]]], [[#References|[5]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Hilbert, "Ueber die Theorie der algebraischen Formen" ''Math. Ann.'' , '''36''' (1890) pp. 473–534 {{MR|1510634}} {{ZBL|22.0133.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Hilbert, "Ueber die vollen Invariantensysteme" ''Math. Ann.'' , '''42''' (1893) pp. 313–373 {{MR|1510781}} {{ZBL|25.0173.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Mumford, "Geometric invariant theory" , Springer (1965) {{MR|0214602}} {{ZBL|0147.39304}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Nagata, "Invariants of a group in an affine ring" ''J. Math. Kyoto Univ.'' , '''3''' (1964) pp. 369–377 {{MR|0179268}} {{ZBL|0146.04501}} </TD></TR></table>