Hilbert's Irreducibility Theorem - Applications

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

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 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.