# SageMath code for working with p-adic field 19.3.3.6a1.3.


# Define the p-adic field: 
Prec = 100  # Default precision of 100
Q19 = Qp(19, Prec); x = polygen(QQ)
L.<t> = Q19.extension(x^3 + 4*x + 17)
K.<a> = L.extension(x^3 + 19)

# Defining polynomial: 
K.defining_polynomial()

# Base field Qp: 
K.base()

# Degree over Qp: 
K.absolute_degree()

# Ramification index: 
K.absolute_e()

# Residue field degree (Inertia degree): 
K.absolute_f()

# Maximal unramified subextension: 
K.maximal_unramified_subextension()

# Roots of unity: 
len(K.roots_of_unity())