Hilbert's irreducibility theorem
   HOME

TheInfoList



OR:

In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, Hilbert's irreducibility theorem, conceived by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
in 1892, states that every finite set of
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted ...
s in a finite number of variables and having
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory.


Formulation of the theorem

Hilbert's irreducibility theorem. Let :f_1(X_1, \ldots, X_r, Y_1, \ldots, Y_s), \ldots, f_n(X_1, \ldots, X_r, Y_1, \ldots, Y_s) be irreducible polynomials in the ring :\Q(X_1, \ldots, X_r) _1, \ldots, Y_s Then there exists an ''r''-tuple of rational numbers (''a''1, ..., ''ar'') such that :f_1(a_1, \ldots, a_r, Y_1,\ldots, Y_s), \ldots, f_n(a_1, \ldots, a_r, Y_1,\ldots, Y_s) are irreducible in the ring :\Q _1,\ldots, Y_s Remarks. * It follows from the theorem that there are infinitely many ''r''-tuples. In fact the set of all irreducible specializations, called Hilbert set, is large in many senses. For example, this set is Zariski dense in \Q^r. * There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (''a''1, ..., ''ar'') to be integers. * There are many
Hilbertian field In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two fundam ...
s, i.e., fields satisfying Hilbert's irreducibility theorem. For example,
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
s are Hilbertian.Lang (1997) p.41 * The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g., it suffices to take n=r=s=1 in the definition. A result of Bary-Soroker shows that for a field ''K'' to be Hilbertian it suffices to consider the case of n=r=s=1 and f=f_1
absolutely irreducible In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.. For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the intege ...
, that is, irreducible in the ring ''K''alg 'X'',''Y'' where ''K''alg is the algebraic closure of ''K''.


Applications

Hilbert's irreducibility theorem has numerous applications in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
and
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
. For example: * The
inverse Galois problem In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There ...
, Hilbert's original motivation. The theorem almost immediately implies that if a finite group ''G'' can be realized as the Galois group of a Galois extension ''N'' of ::E=\Q(X_1, \ldots, X_r), :then it can be specialized to a Galois extension ''N''0 of the rational numbers with ''G'' as its Galois group.Lang (1997) p.42 (To see this, choose a monic irreducible polynomial ''f''(''X''1, ..., ''Xn'', ''Y'') whose root generates ''N'' over ''E''. If ''f''(''a''1, ..., ''an'', ''Y'') is irreducible for some ''ai'', then a root of it will generate the asserted ''N''0.) * Construction of elliptic curves with large rank. * Hilbert's irreducibility theorem is used as a step in the
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awar ...
proof of
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
. * If a polynomial g(x) \in \Z /math> is a perfect square for all large integer values of ''x'', then ''g(x)'' is the square of a polynomial in \Z This follows from Hilbert's irreducibility theorem with n=r=s=1 and ::f_1(X, Y) = Y^2 - g(X). :(More elementary proofs exist.) The same result is true when "square" is replaced by "cube", "fourth power", etc.


Generalizations

It has been reformulated and generalized extensively, by using the language of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. See
thin set (Serre) In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two funda ...
.


References

* D. Hilbert, "Uber die Irreducibilitat ganzer rationaler Functionen mit ganzzahligen Coefficienten", J. reine angew. Math. 110 (1892) 104–129. *{{cite book, first=Serge , last=Lang , authorlink=Serge Lang , title=Survey of Diophantine Geometry , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, year=1997 , isbn=3-540-61223-8 , zbl=0869.11051 *J. P. Serre, ''Lectures on The Mordell-Weil Theorem'', Vieweg, 1989. *M. D. Fried and M. Jarden, ''Field Arithmetic'', Springer-Verlag, Berlin, 2005. *H. Völklein, ''Groups as Galois Groups'', Cambridge University Press, 1996. *G. Malle and B. H. Matzat, ''Inverse Galois Theory'', Springer, 1999. Theorems in number theory Theorems about polynomials David Hilbert