Hilbert's irreducibility theorem

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In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number 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[edit]

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

 \mathbb{Q}[X_1,\ldots, X_r, Y_1,\ldots, Y_s]. \,

Then there exists an r-tuple of rational numbers (a1,...,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

 \mathbb{Q}[Y_1,\ldots, Y_s]. \,

Remarks.

  • It follows from the theorem that there are infinitely many r-tuples. In fact the set of all irreducible specialization, called Hilbert set, is large in many senses. For example, this set is Zariski dense in \mathbb Q^r
  • There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (a1,...,ar) to be integers.
  • 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 recent 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, that is, irreducible in the ring Kalg[X,Y], where Kalg is the algebraic closure of K.

Applications[edit]

Hilbert's irreducibility theorem has numerous applications in number theory and algebra. For example:

  • The inverse Galois problem, 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=\mathbb{Q}(X_1,\ldots, X_r),
then it can be specialized to a Galois extension N0 of the rational numbers with G as its Galois group.[2] (To see this, choose a monic irreducible polynomial f(X1,…,Xn,Y) whose root generates N over E. If f(a1,…,an,Y) is irreducible for some ai, then a root of it will generate the asserted N0.)
  • Construction of elliptic curves with large rank.[2]
  • If a polynomial g(x) \in \mathbb{Z}[x] is a perfect square for all large integer values of x, then g(x) is the square of a polynomial in \mathbb{Z}[x]. 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[edit]

It has been reformulated and generalized extensively, by using the language of algebraic geometry. See thin set (Serre).

References[edit]

  1. ^ Lang (1997) p.41
  2. ^ a b Lang (1997) p.42
  • Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. 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.