In number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, Hilbert's irreducibility theorem, conceived by David Hilbert

David Hilbert

David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

, states that every finite number of irreducible polynomial

Irreducible polynomial

In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....

s in a finite number of variables and having rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a 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

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
  Zariski topology
  In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
  
   in

• 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 field
  Global field
  In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
  
  s are Hilbertian.

• 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 recent result of Bary-Soroker shows that for a field K to be Hilbertian it suffices to consider the case of and absolutely irreducible
  Absolutely irreducible
  In mathematics, absolutely irreducible is a term applied to linear representations or algebraic varieties over a field. It means that the object in question remains irreducible, even after any finite extension of the field of coefficients...
  
  , 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

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

 and algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

. For example:

• The inverse Galois problem
  Inverse Galois problem
  In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers Q. This problem, first posed in the 19th century, is unsolved....
  
  , 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
  Andrew Wiles
  Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory...
  
   proof of Fermat's last theorem
  Fermat's Last Theorem
  In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
  
  .

• 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

It has been reformulated and generalized extensively, by using the language of algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

. See thin set (Serre)

Thin set (Serre)

In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or...

.