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| In [[mathematics]], specifically [[commutative algebra]], '''Hilbert's basis theorem''' states that every [[Ideal (ring theory)|ideal]] in the [[polynomial ring|ring of multivariate polynomials]] over a [[Noetherian ring]] is [[finitely generated module|finitely generated]]. This can be translated into [[algebraic geometry]] as follows: every [[algebraic set]] over a field can be described as the set of common roots of finitely many polynomial equations. {{harvs|txt|authorlink=David Hilbert|last=Hilbert|year=1890}} proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings of invariants.
| | Sculptor Maduena from Shoal Lake, enjoys astrology, diet and hot rods. Has completed a fantastic around the world trip that consisted of taking a trip to the Western Caucasus. |
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| Hilbert produced an innovative proof by contradiction using [[mathematical induction]]; his method does not give an [[algorithm]] to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. One can determine basis polynomials using the method of [[Gröbner basis|Gröbner bases]].
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| ==Proof==
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| <blockquote>'''Theorem.''' If ''R'' is a left (resp. right) [[Noetherian ring]], then the [[polynomial ring]] ''R''[''X''] is also a left (resp. right) Noetherian ring.</blockquote>
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| ''Remark.'' We will give two proofs, in both only the "left" case is considered, the proof for the right case is similar.
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| <br><br />
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| '''First Proof.''' Suppose '''a''' ⊆ ''R''[''X''] were a non-finitely generated left-ideal. Then by recursion (using the [[axiom of countable choice]]) there is a sequence (''f<sub>n</sub>'')<sub>''n''∈'''N</sub>''' of polynomials such that if '''b'''<sub>''n''</sub> is the left ideal generated by ''f''<sub>0</sub>, ..., ''f''<sub>''n''−1</sub> then ''f<sub>n</sub>'' in '''a'''\'''b'''<sub>''n''</sub> is of minimal degree. It is clear that (deg(''f<sub>n</sub>''))<sub>''n''∈'''N</sub>''' is a non-decreasing sequence of naturals. Let ''a<sub>n</sub>'' be the leading coefficient of ''f<sub>n</sub>'' and let '''b''' be the left ideal in ''R'' generated by {''a''<sub>0</sub>, ''a''<sub>1</sub>, ...}. Since ''R'' is left-Noetherian, we have that '''b''' must be finitely generated; and since the ''a<sub>n</sub>'' comprise an ''R''-basis, it follows that for a finite amount of them, say {''a<sub>i</sub>'' : ''i'' < ''N''}, will suffice. So for example, <math>a_N=\sum_{i<N}u_{i}a_{i}\,</math> some ''u<sub>i</sub>'' in ''R''. Now consider
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| :<math>g \triangleq\sum_{i<N}u_{i}X^{\deg(f_{N})-\deg(f_{i})}f_{i},\,</math>
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| whose leading term is equal to that of ''f<sub>N</sub>''; moreover, ''g'' ∈ '''b'''<sub>''N''</sub>. However, ''f<sub>N</sub>'' ∉ '''b'''<sub>''N''</sub>, which means that ''f<sub>N</sub>''−''g'' ∈ '''a'''\'''b'''<sub>''N''</sub> has degree less than ''f<sub>N</sub>'', contradicting the minimality.
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| <br><br />
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| '''Second Proof.''' Let '''a''' ⊆ ''R''[''X''] be a left-ideal. Let '''b''' be the set of leading coefficients of members of '''a'''. This is obviously a left-ideal over ''R'', and so is finitely generated by the leading coefficients of finitely many members of '''a'''; say ''f''<sub>0</sub>, ..., ''f''<sub>''N''−1</sub>. Let <math>d\triangleq\max_{i}\deg(f_{i}).\,</math> Let '''b'''<sub>''k''</sub> be the set of leading coefficients of members of '''a''', whose degree is ≤ ''k''. As before, the '''b'''<sub>''k''</sub> are left-ideals over ''R'', and so are finitely generated by the leading coefficients of finitely many members of '''a''', say <math>f^{(k)}_{0},\ldots,f^{(k)}_{N^{(k)}-1},\,</math> with degrees ≤ ''k''. Now let '''a'''* ⊆ ''R''[''X''] be the left-ideal generated by
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| :<math>\{f_{i},f^{(k)}_{j}:i<N,j<N^{(k)},k<d\}.\,</math>
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| We have '''a'''* ⊆ '''a''' and claim also '''a''' ⊆ '''a'''*. Suppose for the sake of contradiction this is not so. Then let ''h'' ∈ '''a'''\'''a'''* be of minimal degree, and denote its leading coefficient by ''a''.
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| :'''''Case 1:''''' deg(''h'') ≥ ''d''. Regardless of this condition, we have ''a'' ∈ '''b''', so is a left-linear combination <math>a=\sum_j u_j a_j\,</math> of the coefficients of the ''f<sub>j</sub>'' Consider <math>\tilde{h}\triangleq\sum_{j}u_{j}X^{\deg(h)-\deg(f_{j})}f_{j},\,</math> which has the same leading term as ''h''; moreover <math>\tilde{h}\in\mathfrak{a}^{\ast}\not\ni h\,</math> so <math>h-\tilde{h}\in\mathfrak{a}\setminus \mathfrak{a}^{\ast}\,</math> of degree < deg(''h''), contradicting minimality.
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| :'''''Case 2:''''' deg(''h'') = ''k'' < ''d''. Then ''a'' ∈ '''b'''<sub>''k''</sub> so is a left-linear combination <math>a=\sum_j u_j a^{(k)}_j</math> of the leading coefficients of the <math>f^{(k)}_j.</math> Considering <math>\tilde{h}\triangleq\sum_j u_j X^{\deg(h)-\deg(f^{(k)}_{j})}f^{(k)}_{j},</math> we yield a similar contradiction as in ''Case 1''.
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| Thus our claim holds, and '''a''' = '''a'''* which is finitely generated.
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| Note that the only reason we had to split into two cases was to ensure that the powers of ''X'' multiplying the factors, were non-negative in the constructions.
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| == Applications ==
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| Let ''R'' be a Noetherian commutative ring. Hilbert's basis theorem has some immediate corollaries. '''First,''' by induction we see that <math>R[X_{0},X_{1},\ldots,X_{n-1}]\,</math> will also be Noetherian. '''Second,''' since any [[affine variety]] over ''R<sup>n</sup>'' (''i.e.'' a locus-set of a collection of polynomials) may be written as the locus of an ideal <math>\mathfrak{a}\subseteq R[X_{0},X_{1},\ldots,X_{n-1}]\,</math> and further as the locus of its generators, it follows that every affine variety is the locus of finitely many polynomials — i.e. the intersection of finitely many [[hypersurface]]s. '''Finally,''' if <math>\mathcal{A}\,</math> is a finitely-generated ''R''-algebra, then we know that <math>\mathcal{A}\cong R[X_{0},X_{1},\ldots,X_{n-1}]/\langle\mathfrak{a}\rangle\,</math> (''i.e.'' mod-ing out by relations), where '''a''' a set of polynomials. We can assume that '''a''' is an ideal and thus is finitely generated. So <math>\mathcal{A}\,</math> is a free ''R''-algebra (on ''n'' generators) generated by finitely many relations <math>\mathcal{A}\cong R[X_{0},X_{1},\ldots,X_{n-1}]/\langle p_{0},\ldots,p_{N-1}\rangle</math>.
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| ==Mizar System==
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| The [[Mizar system|Mizar project]] has completely formalized and automatically checked a proof of Hilbert's basis theorem in the [http://www.mizar.org/JFM/Vol12/hilbasis.html HILBASIS file].
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| ==References==
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| * Cox, Little, and O'Shea, ''Ideals, Varieties, and Algorithms'', Springer-Verlag, 1997.
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| *{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Ueber die Theorie der algebraischen Formen | doi=10.1007/BF01208503 | year=1890 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=36 | issue=4 | pages=473–534}}
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| [[Category:Commutative algebra]]
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| [[Category:Invariant theory]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in abstract algebra]]
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Sculptor Maduena from Shoal Lake, enjoys astrology, diet and hot rods. Has completed a fantastic around the world trip that consisted of taking a trip to the Western Caucasus.