# 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

${\displaystyle f_{1}(X_{1},\ldots ,X_{r},Y_{1},\ldots ,Y_{s}),\ldots ,f_{n}(X_{1},\ldots ,X_{r},Y_{1},\ldots ,Y_{s})}$

be irreducible polynomials in the ring

${\displaystyle \mathbb {Q} [X_{1},\ldots ,X_{r},Y_{1},\ldots ,Y_{s}].}$

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

${\displaystyle f_{1}(a_{1},\ldots ,a_{r},Y_{1},\ldots ,Y_{s}),\ldots ,f_{n}(a_{1},\ldots ,a_{r},Y_{1},\ldots ,Y_{s})}$

are irreducible in the ring

${\displaystyle \mathbb {Q} [Y_{1},\ldots ,Y_{s}].}$

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 ${\displaystyle \mathbb {Q} ^{r}}$.
• There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (a1,...,ar) to be integers.
• 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 ${\displaystyle n=r=s=1}$ 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 ${\displaystyle n=r=s=1}$ and ${\displaystyle f=f_{1}}$ absolutely irreducible, that is, irreducible in the ring Kalg[X,Y], where Kalg 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
${\displaystyle E=\mathbb {Q} (X_{1},\ldots ,X_{r}),}$
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]
• Hilbert's irreducibility theorem is used as a step in the Andrew Wiles proof of Fermat's last theorem.
• If a polynomial ${\displaystyle g(x)\in \mathbb {Z} [x]}$ is a perfect square for all large integer values of x, then g(x) is the square of a polynomial in ${\displaystyle \mathbb {Z} [x]}$. This follows from Hilbert's irreducibility theorem with ${\displaystyle n=r=s=1}$ and
${\displaystyle f_{1}(X,Y)\,=Y^{2}-g(X)}$.

(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

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.