Yes. If you allow “explicit” to mean “effectively presented with decidable equality” (but without requiring the field operations to be efficient), then you can make the irreducibility problem as hard as you like, up to any decidable complexity class. In particular you can get EXPTIME-hardness (hence, under P ≠ EXPTIME, no polynomial-time algorithm), and you also recover the classical undecidability examples by the same template.

A simple construction

• Start with the purely transcendental field K = Q({t_w : w ∈ {0,1}*}), one independent indeterminate t_w for each binary string w. Equality in K is decidable (it is equality of rational functions over Q in finitely many indeterminates).

• Fix any decidable language L ⊆ {0,1}*. Define the field F_L = K(√t_w : w ∈ L), i.e., adjoin a square root of t_w if and only if w ∈ L.

• Representation and equality: Elements of F_L can be represented as rational functions in finitely many t_u’s together with finitely many adjoined square roots √t_u (with u ∈ L). Normalize elements into the K-basis of the multiquadratic extension determined by the finitely many √t_u that actually occur. Equality in F_L then reduces to equality of finitely many rational functions in Q(t_{u_1}, …, t_{u_k}), which is decidable. Since L is decidable, we can check whether √t_u symbols are legal (u ∈ L). Hence F_L is an explicit field in your sense.

• Hardness via degree-2 polynomials: For each w, consider f_w(X) = X^2 − t_w ∈ F_L[X]. This polynomial is reducible over F_L iff t_w is a square in F_L iff √t_w ∈ F_L iff w ∈ L. Thus the map w ↦ f_w is a many-one reduction from L to the reducibility (equivalently, irreducibility) problem over F_L, using only degree-2 inputs with two coefficients (1 and −t_w). Encoding t_w by the string w makes the reduction polynomial-time.

Consequences

• Choose L to be EXPTIME-complete. Then irreducibility (and hence factorization) over F_L[X] is EXPTIME-hard under polynomial-time reductions. Therefore there is no polynomial-time algorithm unless P = EXPTIME. In practice, the best known general algorithm will be exponential-time in the worst case (since even deciding reducibility of the special inputs f_w requires solving an EXPTIME-hard problem).

• More generally, for any decidable time class C (e.g., DTIME(T(n)) for any time-constructible T), you can make irreducibility over F_L be C-hard. In particular, by the deterministic time hierarchy theorem, you can pick L ∈ EXPTIME \ P to get an unconditional superpolynomial lower bound (no polynomial-time algorithm) for irreducibility over F_L.

• If you take L undecidable, you recover Fröhlich–Shepherdson-style undecidability: irreducibility over F_L becomes undecidable. The same construction shows that (without further restrictions on the ground field presentation) there is no uniform upper bound phenomenon beyond what you build into L.

Remarks

• This hardness already appears on degree-2 polynomials, so the factorization problem inherits the same lower bounds.

• If you additionally require that the ground field operations (and equality) be efficiently computable (e.g., polynomial-time in the bitlength of the inputs), then all known “natural” fields (Q, fixed number fields, finite fields, function fields over finite fields) admit polynomial-time factorization. Producing exponential-time lower bounds in that more constrained setting would require major breakthroughs (e.g., proving concrete time lower bounds for natural problems). The construction above shows that, absent such efficiency constraints on the field presentation, one can manufacture explicit fields with arbitrarily hard irreducibility.