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| {{Hatnote|For the theorem about the real analytic Eisenstein series see [[Kronecker limit formula]].}}
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| In [[mathematics]], '''Kronecker's theorem''' is either of two theorems named after [[Leopold Kronecker]].
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| __NOTOC__
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| == The existence of extension fields ==
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| This is a theorem stating that a [[polynomial]] in a [[Field (mathematics)|field]], ''p''(''x'') ∈ ''F''[''x''], has a root in an extension field <math>E \supset F</math>.<ref>[http://www.usna.edu/Users/math/wdj/book/node77.html Applied Abstract Algebra] by D. Joyner, R. Kreminski and J. Turisco.</ref>
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| For example, a polynomial in the [[Real number|reals]] such as ''x''<sup>2</sup> + 1 = 0 has two roots, both in the complex field.
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| This theorem is usually credited to Kronecker despite his original reluctance to accept the existence of numbers outside of the rationals;<ref>{{Cite book | last=Allenby | first=R. B. J. T.| title=Rings, fields and groups: an introduction to abstract algebra | date=1983 | publisher=E. Arnold | location=London | isbn=0-7131-3476-3 | pages=140,141}}</ref> it provides a useful [[Constructivism (mathematics)|construction]] of many sets.
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| == A result in diophantine approximation ==
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| '''Kronecker's theorem''' may also refer to a result in [[diophantine approximation]]s applying to several [[real number]]s ''x<sub>i''</sub>, for 1 ≤ ''i'' ≤ ''N'', that generalises [[Dirichlet's approximation theorem]] to multiple variables. In terms of physical systems, it has the consequence that planets in circular orbits moving uniformly around a star will, over time, assume all alignments, unless there is an exact dependency between their orbital periods.
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| In the case of ''N'' numbers, taken as a single ''N''-[[tuple]] and point ''P'' of the [[torus]]
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| :''T'' = ''R<sup>N</sup>/Z<sup>N</sup>'', | |
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| the [[closure (mathematics)|closure]] of the subgroup <''P''> generated by ''P'' will be finite, or some torus ''T′'' contained in ''T''. The original '''Kronecker's theorem''' ([[Leopold Kronecker]], 1884) stated that the [[necessary condition]] for
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| :''T′'' = ''T'', | |
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| which is that the numbers ''x<sub>i''</sub> together with 1 should be [[linearly independent]] over the [[rational number]]s, is also [[sufficient condition|sufficient]]. Here it is easy to see that if some [[linear combination]] of the ''x<sub>i''</sub> and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a [[character (mathematics)|character]] χ of the group ''T'' other than the [[trivial character]] takes the value 1 on ''P''. By [[Pontryagin duality]] we have ''T′'' contained in the [[Kernel (group theory)|kernel]] of χ, and therefore not equal to ''T''.
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| In fact a thorough use of Pontryagin duality here shows that the whole Kronecker theorem describes the closure of <''P''> as the intersection of the kernels of the χ with
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| :χ(''P'') = 1.
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| This gives an ([[antitone]]) [[Galois connection]] between [[Monogenic semigroup|monogenic]] closed subgroups of ''T'' (those with a single generator, in the topological sense), and sets of characters with kernel containing a given point. Not all closed subgroups occur as monogenic; for example a subgroup that has a torus of dimension ≥ 1 as connected component of the identity element, and that is not connected, cannot be such a subgroup.
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| The theorem leaves open the question of how well (uniformly) the multiples ''mP'' of ''P'' fill up the closure. In the one-dimensional case, the distribution is uniform by the equidistribution theorem.
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| ==See also==
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| * [[Kronecker set]]
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| * [[Weyl's criterion]]
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| == Notes and references ==
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| {{Springer|id=k/k055910|title=Kronecker's theorem}}
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| <references/>
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| [[Category:Diophantine approximation]]
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| [[Category:Topological groups]]
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| [[Category:Theorems in abstract algebra]]
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