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In mathematics, Kronecker's theorem is either of two theorems named after Leopold Kronecker.
The existence of extension fields

This is a theorem stating that a polynomial in a field, p(x) ∈ F[x], has a root in an extension field $$E \supset F$$.

For example, a polynomial in the reals such as x2 + 1 = 0 has two roots, both in the complex field.

This theorem is usually credited to Kronecker despite his original reluctance to accept the existence of numbers outside of the rationals; it provides a useful construction of many sets.
A result in diophantine approximation

Kronecker's theorem may also refer to a result in diophantine approximations applying to several real numbers xi, for 1 ≤ i ≤ N, that generalises[dubious – discuss] the equidistribution theorem, which implies that an infinite cyclic subgroup of the unit circle group is a dense subset. 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.

In the case of N numbers, taken as a single N-tuple and point P of the torus

T = RN/ZN,

the 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

T′ = T,

which is that the numbers xi together with 1 should be linearly independent over the rational numbers, is also sufficient. Here it is easy to see that if some linear combination of the xi and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a 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 of χ, and therefore not equal to T.

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

χ(P) = 1.

This gives an (antitone) Galois connection between 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.

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.

Kronecker set
Weyl's criterion

Notes and references

Hazewinkel, Michiel, ed. (2001), "Kronecker's theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104

^ Applied Abstract Algebra by D. Joyner, R. Kreminski and J. Turisco.
^ Allenby, R. B. J. T. (1983). Rings, fields and groups: an introduction to abstract algebra. London: E. Arnold. pp. 140,141. ISBN 0-7131-3476-3.

Mathematics Encyclopedia