The reduction modulo two is reducible, since x^4+1=(x+1)^4 there, so it cannot support the published irreducibility implication. Modulo eleven, Rabin's degree-four test succeeds: the p^4 remainder is x and the gcd of f with x^(p^2)-x is one. Hence the primitive polynomial is irreducible over the rationals. This repairs only the irreducibility step and does not verify the later Galois-group or density claims.
RESULT_JSON:{"bad_prime":2,"bad_reduction":[1,0,0,0,1],"bad_factor":[1,1],"bad_factor_power":4,"repair_prime":11,"p2_remainder":[10,9,6,4],"p4_remainder":[0,1],"rabin_gcd_degree":0,"rational_conclusion":"IRREDUCIBLE_OVER_Q"}
