Hilbert's theorem of irreducibility

The Irreduzibilitätssatz of Hilbert is a set of David Hilbert on the irreducibility of polynomials with rational coefficients in several variables, when a number of variables are rational values. Generalizations of the set of polynomials relating to other bodies than the rational numbers. The theorem is of particular importance for number theory and arithmetic algebraic geometry .

Formulation of the sentence

Hilbert's theorem of irreducibility reads: Let

${\displaystyle f(x_{1}\ldots ,x_{s},y)}$

an irreducible polynomial over the rational numbers. Then there are infinitely many tuples of rational numbers such that: ${\displaystyle s}$ ${\displaystyle q_{i}}$

${\displaystyle f(q_{1},\ldots ,q_{s},y)}$

is irreducible.

example

${\displaystyle f(x,y)=y^{2}-x}$remains irreducible over the rational numbers for all specializations that are not squares of rational numbers. One has on the other hand for that the expression reducible rational for all , that is a square in the rational numbers, the original expression can after Irreduzibilitätssatz not irreducible over the rational numbers and be a polynomial has the square be . The same applies to with and when rational is replaced by an integer. ${\displaystyle x=q}$ ${\displaystyle f(x_{1},...,x_{i},y)=y^{2}-g(x_{1},...,x_{i})}$ ${\displaystyle (x_{1},...,x_{i})}$ ${\displaystyle g(x_{1},...,x_{i})}$ ${\displaystyle g(x)=a(x)^{2}}$ ${\displaystyle y^{n}-g(x)}$ ${\displaystyle n>2}$

Hilbert body

In a more general way, one can also consider bodies other than the rational numbers, if Hilbert's law of irreducibility applies to them, one speaks of Hilbert bodies . In addition to the rational numbers, examples of Hilbert fields are algebraic extensions of the rational numbers, i.e. algebraic number fields . The irreducibility theorem also applies to specializations in the whole numbers and the rings of whole numbers in algebraic number fields. The irreducibility theorem ensures the existence of an infinite number of such s-tuples of numbers ; one can even show that they are Zariski - close in . The theorem has applications in number theory, for example it is used in the proof of the great Fermat conjecture by Andrew Wiles , and in the inverse Galois theory . ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k^{s}}$

Applications

The irreducibility theorem was used by André Neron for the construction of Abelian varieties over the rational numbers. According to Mordell-Weil's theorem , the group of rational points of the Abelian variety is finitely generated (and the rank finite) and Neron constructed Abelian varieties of dimension g and rank . In 1952 Neron showed that there are elliptic curves with rank above the rational numbers. ${\displaystyle r\geq 3g+6}$ ${\displaystyle r>9}$

David Hilbert and Emmy Noether (1918) used the theorem in the inverse Galois theory. This is about the problem of finding a field extension L of k such that a given finite group G is its Galois group.

Introduce a variable for each group element and consider the effect of G on the permutation of the variables , giving a representation of G in , where ) is the group order . Let be the subfield of K invariant under G, which in turn is a subfield of the field of symmetric functions in the n variables over K and is a rational function field over k. Then the Galois group G. If k is a Hilbert field, for example , one can specify (n-1) of the variables and according to the irreducibility theorem we get a field L which is the decay field of a polynomial and has the Galois group G. ${\displaystyle g\in G}$ ${\displaystyle x_{g}}$ ${\displaystyle h(x_{g})=x_{hg}}$ ${\displaystyle K=k(x_{1},...,x_{n})}$ ${\displaystyle n=|G|}$ ${\displaystyle K^{G}}$ ${\displaystyle x_{i}}$ ${\displaystyle K\backslash K^{G}}$ ${\displaystyle k=\mathbb {Q} }$ ${\displaystyle f(x)\in k(x)}$

This strategy by Emmy Noether (Noether's criterion or Noether's conjecture, the invariant field of G is a rational function field over k, that is, a purely transcendent field expansion of k) is often successful. Noether showed this for the case of the symmetric group and other examples are known, such as Claude Chevalley for finite reflection groups. Richard Swan found a counterexample in 1969 with the cyclic group of order 47. The inverse problem of Galois theory, formulated by Hilbert in 1892, is generally unsolved to this day, even if many partial results are known. For example, every finite Abelian group can be represented as a Galois group over ( Kronecker-Weber theorem ), and likewise every finite group that can be resolved ( Igor Schafarewitsch ). ${\displaystyle \mathbb {Q} }$

literature

• David Hilbert, On the irreducibility of whole rational functions with integer coefficients, Journal für Reine und Angewandte Mathematik, Volume 110, 1892, pp. 104–129, SUB Göttingen
• Jean-Pierre Serre: Topics in Galois Theory, Jones and Bartlett 1992 (Chapter 3)
• Charles Robert Hadlock: Field theory and its classical problems, Carus Mathematical Monographs, Mathematical Association of America 1978
• Serge Lang: Fundamentals of Diophantine Geometry, Springer 1983 (or Diophantine Geometry, Interscience 1962, Chapter 8)

Web links

• Umberto Zannier , On the Hilbert irreducibility theorem, Rend. Mat. Sem. Univ. Pole. Torino, Volume 67, 2009, pp. 1-14, pdf
• Alexei Nikolajewitsch Parschin , Hilbert's irreducibility theorem, Encyclopedia of Mathematics, Springer

Individual evidence

1. ^ Lang, Survey of Diophantine Geometry, Springer 1997, p. 41
2. ↑ Parshin, in: Hilbert theorems, Encyclopedia of Mathematical Sciences, Springer
3. ^ Emmy Noether, equations with a prescribed group, Mathematische Annalen, Volume 78, 1918, pp. 221–229, SUB Göttingen
4. ↑ Meredith Blue, Galois theory and Noether's problem, Proc. Thirty-Fourth Annual Meeting Florida Section MAA, 2001, pdf
5. ^ For Hilbert, G was the symmetric group, which Noether generalized.
6. ^ Matzat, Konstruktiv Galoistheorie, Springer, 1987, p. 5
7. ^ Jacques Martinet Un contre-exemple à une conjecture d'E. Noether, d'après R. Swan, Seminaire Bourbaki 372, 1969, numdam ( Memento of the original from March 3, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.