// Magma code for working with p-adic field 5.2.2.2a1.1.


// Define the p-adic field: 
Prec := 100;  // Default precision of 100
Q5 := pAdicField(5, Prec);
K := LocalField(Q5, Polynomial(Q5, [4, 21, 20, 8, 1]));

// Defining polynomial: 
DefiningPolynomial(K);

// Base field Qp: 
Q5;

// Degree over Qp: 
Degree(K);

// Ramification index: 
RamificationIndex(K);

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

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