// Magma code for working with p-adic field 3.10.2.10a1.2.


// Define the p-adic field: 
Prec := 100;  // Default precision of 100
Q3 := pAdicField(3, Prec);
K := LocalField(Q3, Polynomial(Q3, [7, 4, 1, 0, 8, 12, 12, 4, 4, 8, 16, 10, 4, 0, 4, 4, 4, 0, 0, 0, 1]));

// Defining polynomial: 
DefiningPolynomial(K);

// Base field Qp: 
Q3;

// Degree over Qp: 
Degree(K);

// Ramification index: 
RamificationIndex(K);

// Residue field degree (Inertia degree): 
InertiaDegree(K);

// Discriminant exponent: 
Valuation(Discriminant(K));