Chebotaryov density theorem

The Chebotaryov density theorem (depending on the transcription also the density theorem by Chebotarëv or Tschebotareff ) is a generalization of Dirichlet's theorem about prime numbers in arithmetic progressions to Galois extensions of number fields . In the case of abelian extension of it is obtained back to the proposition that the amount of the primes of the form , natural-tightness has. In its general form, it follows in particular the theorem, proved by Kronecker in 1880 , that precisely the prime numbers are completely decomposed in a given Galois expansion of of degree . ${\displaystyle \mathbb {Q} }$ ${\displaystyle ak+n}$ ${\displaystyle (a,n)=1}$ ${\displaystyle 1/\phi (n)}$ ${\displaystyle 1/n}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle n}$

The sentence was found by Nikolai Grigoryevich Chebotaryov in 1922 and first published in Russian in 1923 and in German in 1925.

formulation

Let be a Galois extension of number fields with , and a conjugation class . Then the set of unbranched prime ideals of whose Frobenius element (in the case of a non-Abelian extension this is generally a conjugation class) has natural density ${\displaystyle L|K}$ ${\displaystyle G=G(L|K)}$ ${\displaystyle C\subset G}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle K}$ ${\displaystyle C}$

${\displaystyle {\frac {|C|}{|G|}}}$.

Applications

For an Abelian extension, for example in the case of quadratic number fields , each conjugation class consists of exactly one element, which is why a uniform distribution is obtained. Is the non-Abelian group of order , so there are the conjugation of 1, 3 and 2 members, so that the primes of three primes fully decomposed , in exactly two decomposed (with degree of inertia and ) and sluggish are. ${\displaystyle G=S_{3}}$ ${\displaystyle 6}$ ${\displaystyle \{\mathrm {id} \},\{(1,2),(1,3),(2,3)\},\{(1,2,3),(1,3,2)\}}$ ${\displaystyle 1/6}$ ${\displaystyle K}$ ${\displaystyle 3/6=1/2}$ ${\displaystyle 2}$ ${\displaystyle 1}$ ${\displaystyle 2/6=1/3}$

One can also deduce from this that for compound numbers there is an irreducible polynomial over a global field , so that it is reducible over all local completions . For example, this applies to everyone with a Galois group isomorphic to the Klein group of four . ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle K}$ ${\displaystyle f}$ ${\displaystyle K_{\mathfrak {p}}}$ ${\displaystyle f\in \mathbb {Z} [X]}$ ${\displaystyle C_{2}\times C_{2}}$

By decomposing a polynomial into remainder class fields, one can also obtain information about the structure of its Galois group and limit this probabilistically with Chebotaryov's density theorem.

If modulo almost all prime numbers completely decompose into linear factors, it also decomposes completely over; this is a kind of local-global principle . If an irreducible polynomial with integer coefficients that has a zero modulo almost all prime numbers, it has degrees . ${\displaystyle f\in \mathbb {Z} [X]\setminus \{0\}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle f}$ ${\displaystyle 1}$

If Galois extensions of a number field and if the set of prime ideals of , which are divided into or fully divided, is the same apart from finitely many exceptions, then it follows . (The prerequisite that the extensions are Galois can not be dropped.) A Galois extension is therefore uniquely determined by the set of fully decomposed prime ideals. In order to classify the Galois extensions of , it suffices to determine the sets of prime ideals of , which can appear as sets of fully decomposed prime ideals. For Abelian extensions this happens precisely through the class field theory ; for non-Abelian extensions this is still an unsolved problem, see Langlands program . ${\displaystyle L,M}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle M}$ ${\displaystyle L=M}$ ${\displaystyle K}$ ${\displaystyle K}$

literature

• P. Stevenhagen, HW Lenstra: Chebotarëv and his Density Theorem. In: The Mathematical Intelligencer . Vol. 18, No. 2, 1996, pp. 26-37 ( PDF; 2.7 MB ).
• N. Tschebotareff: The determination of the density of a set of prime numbers which belong to a given substitution class. In: Mathematical Annals . Vol. 95, No. 1, 1925, pp. 191-228 ( digitized version ).

Individual evidence

1. ^ Robert Guralnick, Murray M. Schacher, Jack Sonn: Irreducible polynomials which are locally reducible everywhere. In: Proceedings of the American Mathematical Society. Vol. 133, 2005, ISSN  0002-9939 , pp. 3171-3177 ( digitized version ).