I am interested in generating irreducible polynomials of a given, arbitrary degree over either the reals or rationals using integer coefficients. They don't necessarily have to be arbitrary polynomials (i.e. random coefficients), but I would prefer not to simply use $F(x)=x^n+a_n$ (for the case of $\mathbb{R}$.
If a general algorithm for generating arbitrary irreducible polynomials does not exist, I am open to using several "families" of polynomials of arbitrary degree which are known to be irreducible, in which case I would like to be able to generate several different such families.
Over the rationals, use the Eisenstein Irreducibility Criterion.
Over the reals is not interesting, no polynomial of degree $\ge 3$ is irreducible.