# Hilbert's irreducibility theorem

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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

[edit]**Hilbert's irreducibility theorem.** Let

be irreducible polynomials in the ring

Then there exists an *r*-tuple of rational numbers (*a*_{1}, ..., *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 specializations, called Hilbert set, is large in many senses. For example, this set is Zariski dense in - There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (
*a*_{1}, ...,*a*) to be integers._{r} - There are many Hilbertian fields, i.e., fields satisfying Hilbert's irreducibility theorem. For example, number 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 in the definition. A result of Bary-Soroker shows that for a field
*K*to be Hilbertian it suffices to consider the case of and absolutely irreducible, that is, irreducible in the ring*K*^{alg}[*X*,*Y*], where*K*^{alg}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

- then it can be specialized to a Galois extension
*N*_{0}of the rational numbers with*G*as its Galois group.^{[2]}(To see this, choose a monic irreducible polynomial*f*(*X*_{1}, ...,*X*,_{n}*Y*) whose root generates*N*over*E*. If*f*(*a*_{1}, ...,*a*,_{n}*Y*) is irreducible for some*a*, then a root of it will generate the asserted_{i}*N*_{0}.)

- 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 is a perfect square for all large integer values of
*x*, then*g(x)*is the square of a polynomial in This follows from Hilbert's irreducibility theorem with and

- (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]- D. Hilbert, "Uber die Irreducibilitat ganzer rationaler Functionen mit ganzzahligen Coefficienten", J. reine angew. Math. 110 (1892) 104–129.

- 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.