# Kronecker's theorem

For the theorem about the real analytic Eisenstein series see Kronecker limit formula.

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 non-constant polynomial in a field, p(x) ∈ F[x], has a root in an extension field $E \supset F$.[1]

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;[2] 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 ≤ in, that generalises Dirichlet's approximation theorem to multiple variables.

The classical Kronecker's approximation theorem is formulated as follows; Given real numbers $\alpha_i=(\alpha_{i_1},\cdots,\alpha_{i_n})\in\mathbb{R}^n, i=1,\cdots,m$ and $\beta_j=(\beta_1,\cdots,\beta_n)\in \mathbb{R}^n$ , for any small $\epsilon>0$ there exist integers $p_i$ and $q_j$ such that

$\biggl| \sum^m_{i=1}q_i\alpha_{ij}-p_j-\beta_j\biggr|<\epsilon,\ \ \ \ 1\le j\le n$ ,

if and only if for any $r_1,\dots,r_n\in\mathbb{Z},\ i=1,\dots,m$ with

$\sum^n_{j=1}\alpha_{ij}r_j\in\mathbb{Z}, \ \ i=1,\dots,m\ ,$

the number $\sum^n_{j=1}\beta_jr_j$ is also an integer.

Kronecker's approximation theorem had been firstly proved by L. Kronecker in the end of the 19-th century. It has been now revealed to relate to the idea of n-torus and Mahler measure since the later half of the 20-th century. 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.

### the relation to n-torus

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.