Hilbert's irreducibility theorem

From Wikipedia, the free encyclopedia

In number theory a subject of mathematics, 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.

Contents 1 Formulation of the Theorem 2 Applications 3 Generalizations 4 References

[edit] Formulation of the Theorem

Hilbert's Irreducibility Theorem. Let

be irreducible polynomials in the ring

Then there exists an r-tuple of rational numbers (a₁,...,a_r) such that

are irreducible in the ring

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

• There are always integer specialization, i.e., the assertion of the theorem holds even if we demand (a₁,...,a_r) 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 Hilebrtian it suffices to consider the case of n = r = s = 1 and f = f₁ absolutely irreducible, that is, irreducible in the ring K^alg[X,Y], where K^alg is the algebraic closure of K.

• There are many Hilbertian fields, i.e., fields satisfying Hilbert's irreducibility theorem. For example, global fields are Hilbertian.

[edit] Applications

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

then it can be specialized to a Galois extension N₀ of the rational numbers with G as its Galois group. (To see this, choose a monic irreducible polynomial f(X₁,…,X_n,Y) whose root generates N over E. If f(a₁,…,a_n,Y) is irreducible for some a_i, then a root of it will generate the asserted N₀.)

• Construction of elliptic curves with large rank.

• Hilbert's irreducibility theorem is used as a step in the Andrew Wiles proof of Fermat's last theorem.

[edit] Generalizations

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

[edit] References

• 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.