Are Irreducible over the Rationals?

When the quadratic factors have no rational roots, only irrational roots involving radicals or complex numbers, then it is said to be irreducible over the rationals.

Which polynomials are irreducible over Q?

Definition 1. A given monic polynomial with integral coefficients has the property (И) if it is irreducible over Q but reducible over Fp for every prime Ф. theorem, linearly disjoint extensions, linear relations connecting roots of polynomials.

How do you know if a polynomial is irreducible?

If a polynomial with degree 2 or higher is irreducible in , then it has no roots in . If a polynomial with degree 2 or 3 has no roots in , then it is irreducible in .