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Hilbert Irreducibility Theorem

The Hilbert irreducibility theorem states that if f(t,x) in Q(t)[x] is an irreducible polynomial that has positive polynomial degree in x, then there are infinitely many a in Q for which the specialized polynomial f(a,x) in Q[x] is defined and remains an irreducible polynomial.

One consequence is that if G!={1} is a finite group and L/Q(t) is a regular Galois extension whose Galois group is isomorphic to G, then infinitely many specializations give G-extensions of Q.

See also

Galois Extension Field, Galois Group, Inverse Galois Problem, Regular Galois Extension

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References

Malle, G. and Matzat, B. H. Inverse Galois Theory, 2nd ed. Berlin, Germany: Springer, 2018.

Cite this as:

Weisstein, Eric W. "Hilbert Irreducibility Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HilbertIrreducibilityTheorem.html

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