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For quadratic fields the situation is simpler than the general case. I see no reason why prime ideals should not have (easily implemented) methods is_inert(), is_split(), though in the latter case it is not so clear what is meant for non-Galois extensions. However, prime ideals do know what their residue degree is and their ramification degree (called ramification_index for some reason) so you can do this:

sage: K.<a> = QuadraticField(5)
sage: [[(p,P.residue_class_degree(), P.ramification_index()) for P in K.primes_above(p)] for p in prime_range(50)]
[[(2, 2, 1)],
 [(3, 2, 1)],
 [(5, 1, 2)],
 [(7, 2, 1)],
 [(11, 1, 1), (11, 1, 1)],
 [(13, 2, 1)],
 [(17, 2, 1)],
 [(19, 1, 1), (19, 1, 1)],
 [(23, 2, 1)],
 [(29, 1, 1), (29, 1, 1)],
 [(31, 1, 1), (31, 1, 1)],
 [(37, 2, 1)],
 [(41, 1, 1), (41, 1, 1)],
 [(43, 2, 1)],
 [(47, 2, 1)]]

So ramified prime look like (p,1,2), intert ones (p,2,1) and split ones (p,1,1).