Hilbert's irreducibility theorem

In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set 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

:<math>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) </math>

be irreducible polynomials in the ring

:<math>\Q(X_1, \ldots, X_r)[Y_1, \ldots, Y_s].</math>

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

:<math>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)</math>

are irreducible in the ring

:<math>\Q[Y_1,\ldots, Y_s].</math>

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 <math>\Q^r.</math>
• 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, number fields are Hilbertian.
• The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g., it suffices to take <math>n=r=s=1</math> in the definition. A result of Bary-Soroker shows that for a field K to be Hilbertian it suffices to consider the case of <math>n=r=s=1</math> and <math>f=f_1</math> 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

::<math>E=\Q(X_1, \ldots, X_r),</math>

: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.