sage: K = CyclotomicField(23)
sage: def ef(q):
....:   # takes a prime ideal q and returns the ramification and inertia indices of q
....:   # over the prime ideal p of the base ring lying under q
....:   return q.ramification_index(), q.residue_class_degree()
....: 
sage: ef(K.factor(23)[0][0])
(22, 1)
sage: ef(K.factor(2)[0][0])
(1, 11)
sage: ef(K.factor(2)[1][0])
(1, 11)
sage: def splitp(K, p):
....:   # lists the prime factors of p O_K with their e and f
....:   return [(x[0], x[0].ramification_index(), x[0].residue_class_degree()) for x in K.factor(p)]
....: 
sage: splitp(K, 23)
[(Fractional ideal (23, zeta23 - 1), 22, 1)]
sage: splitp(K, 2)
[(Fractional ideal (2, zeta23^11 + zeta23^9 + zeta23^7 + zeta23^6 + zeta23^5 + zeta23 + 1),
  1,
  11),
 (Fractional ideal (2, zeta23^11 + zeta23^10 + zeta23^6 + zeta23^5 + zeta23^4 + zeta23^2 + 1),
  1,
  11)]
sage: splitp(K, 3)
[(Fractional ideal (3, zeta23^11 - zeta23^8 - zeta23^6 + zeta23^4 + zeta23^3 - zeta23^2 - zeta23 - 1),
  1,
  11),
 (Fractional ideal (3, zeta23^11 + zeta23^10 + zeta23^9 - zeta23^8 - zeta23^7 + zeta23^5 + zeta23^3 - 1),
  1,
  11)]
sage: splitp(K, 5)
[(Fractional ideal (5), 1, 22)]
sage: splitp(K, 47)
[(Fractional ideal (47, zeta23 + 10), 1, 1),
 (Fractional ideal (47, zeta23 + 11), 1, 1),
 (Fractional ideal (47, zeta23 + 13), 1, 1),
 (Fractional ideal (47, zeta23 + 15), 1, 1),
 (Fractional ideal (47, zeta23 + 19), 1, 1),
 (Fractional ideal (47, zeta23 + 20), 1, 1),
 (Fractional ideal (47, zeta23 + 22), 1, 1),
 (Fractional ideal (47, zeta23 + 23), 1, 1),
 (Fractional ideal (47, zeta23 - 21), 1, 1),
 (Fractional ideal (47, zeta23 - 18), 1, 1),
 (Fractional ideal (47, zeta23 - 17), 1, 1),
 (Fractional ideal (47, zeta23 - 16), 1, 1),
 (Fractional ideal (47, zeta23 - 14), 1, 1),
 (Fractional ideal (47, zeta23 - 12), 1, 1),
 (Fractional ideal (47, zeta23 - 9), 1, 1),
 (Fractional ideal (47, zeta23 - 8), 1, 1),
 (Fractional ideal (47, zeta23 - 7), 1, 1),
 (Fractional ideal (47, zeta23 - 6), 1, 1),
 (Fractional ideal (47, zeta23 - 4), 1, 1),
 (Fractional ideal (47, zeta23 - 3), 1, 1),
 (Fractional ideal (47, zeta23 - 2), 1, 1),
 (Fractional ideal (47, zeta23 + 5), 1, 1)]
sage: splitp(QQ[2^(1/3)], 2)
[(Fractional ideal (a), 3, 1)]
sage: splitp(QQ[2^(1/3)], 3)
[(Fractional ideal (a + 1), 3, 1)]
sage: splitp(QQ[2^(1/3)], 5)
[(Fractional ideal (a^2 - 2*a - 1), 1, 2), (Fractional ideal (-a^2 - 1), 1, 1)]
sage: