Lecture 2 : The profinite ring Z_p of p-adic integers, p-adic expansion, multiplication by p, units, the p-adic valuation, the p-adic distance

The field of Q_p of p-adic numbers, squares and quadratic characters of Q_p^*, structure of the group Q_p^*.

The Hilbert symbol, Steinberg property, explicit formulae, the conic ax^2+by^2=1.

Lecture 8 : Power series over a finite field F_p, analogy with Z_p, the field of fractions F_p((t)), squares in F_p((t))^*, quadratic characters of F_p((t))^*, the Hilbert symbol, Steinberg property, explicit formulae, the conic ax^2+by^2=1, quadratic extensions of F_p((t)), bijection with the set of quadratic characters.

Lecture 10 : Structure of the filtered space k((t))^* modulo p-th powers, structure of k((t))^+ modulo the Artin-Schreier map (k finite field), the snake lemma.

Lecture 12 : Structure theorem for p-fields, ramification index, residual degree, raising to the power p in p-fields of characteristic 0, the filtration on the quotient modulo p-th powers.

Lecture 14 : Hensel's lemma, extension of a complete discrete valuation to a finite extension field, integral closure of the ring of integers.

Lecture 16 : Relation between the degree, the ramification index, and the residual degree, the norm map, monogenity of the ring of integers in a finite extension, unramified extensions, totally ramified extensions, tamely ramified extensions.

Lecture 18 : Maximal tamely ramified extension, galoisian and abelian tamely ramified extensions, maximal tamely ramified abelian extension, the number of tamely ramified extensions of various kinds.

Lecture 20 : Inertia subgroup and ramification subgroup, higher ramification groups, lower numbering, ramification filtration on a subgroup.

Lecture 22 : Ramification filtration on cyclotomic extensions, ramification jumps, cyclic extensions, abelian extensions of prime exponent, Hasse-Arf.

Lecture 24 : ramification breaks of cyclic extensions in the absence of a primitive p-th root of unity, contribution of cyclic extensions to Serre's mass formula.

Lecture 26 : Specialisation to the case of p-fields, the maximal abelian extension of exponent dividing p-1, the galois module structure of units modulo p-th powers, Serre's mass formula in prime degree.

Lecture 28 : The local Kronecker-Weber theorem.

Lecture 30: Primitive solvable extensions of an arbitrary field.

Lectures 32 : Irreducible F_p-representations of the absolute Galois group of a p-field.

Lectures 34 : The Galois module structure of units modulo p-th powers in a tamely ramified split extension.

Lecture 36 : Primitive extensions of p-fields.