Hilbert's irreducibility theorem

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

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,...,a_r) to be integers.

There are many Hilbertian fields, i.e., fields satisfying Hilbert's irreducibility theorem. For example, global fields are Hilbertian.[1]

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 K^alg[X,Y], where K^alg is the algebraic closure of K.

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

\( E=\mathbb{Q}(X_1,\ldots, X_r), \)

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,…,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.[2]

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

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

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

References

Lang (1997) p.41

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.

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