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Are the ramified primes the ones which divide the field discriminant? If this is so you can get them by

sage: K.<y> = NumberField(x^4 - 420*x^2 + 40000)
sage: [x[0] for x in list(K.discriminant().factor())]
[5, 41]

Are the ramified primes the ones which divide the field discriminant? If this is so you can get them by

sage: K.<y> = NumberField(x^4 - 420*x^2 x^2 + 40000)
1)
sage: [x[0] for x in list(K.discriminant().factor())]
[5, 41]
[2, 3]

If splitting means that the prime factors then you can check this like this:

sage: is_split = lambda F,x:sum([t[1] for t in list(F.factor(x))])>1

for example:

sage: K.<y> = NumberField(x^2 + 1)
sage: for x in range(30):
    if is_prime(x):
        print x%4,is_split(K,x)
....:         
2 True
3 False
1 True
3 False
3 False
1 True
1 True
3 False
3 False
1 True