Kronecker theorem
Given ,
, and
; then for any
there exist integers
,
, and
,
, such that
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if and only if for any such that
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the number
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is also an integer. This theorem was first proved in 1884 by L. Kronecker (see [1]).
Kronecker's theorem is a special case of the following theorem [2], which describes the closure of the subgroup of the torus generated by the elements
,
: The closure is precisely the set of all classes
such that, for any numbers
with
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one has also
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(Cf. [2].) Under the assumptions of Kronecker's theorem, this closure is simply . This means that the subgroup of all elements of the form
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where , is dense in
, while the subgroup of vectors
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where , is dense in
. Kronecker's theorem can be derived from the duality theory for commutative topological groups (cf. Topological group), [3].
In the case , Kronecker's theorem becomes the following proposition: A class
, where
, generates
as a topological group if and only if the numbers
are linearly independent over the field
of rational numbers. In particular, the torus
as a topological group is monothetic, i.e. is generated by a single element.
References
[1] | L. Kronecker, "Näherungsweise ganzzahlige Auflösung linearer Gleichungen" , Werke , 3 , Chelsea, reprint (1968) pp. 47–109 |
[2] | N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) |
[3] | L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) |
Comments
The last statement above can be rephrased as: If are linearly independent over
, then the set
is dense in
. Here
denotes the fractional part of
(cf. Fractional part of a number). In fact, the set
is even uniformly distributed, cf. Uniform distribution.
References
[a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. 23 |
[a2] | J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957) |
Kronecker theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kronecker_theorem&oldid=19181