# SageMath code for working with p-adic field 7.12.1.0a1.1.


# Define the p-adic field: 
Prec = 100  # Default precision of 100
Q7 = Qp(7, Prec); x = polygen(QQ)
K.<a> = Q7.extension(x^12 + 2*x^8 + 5*x^7 + 3*x^6 + 2*x^5 + 4*x^4 + 5*x^2 + 3)

# 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())