Convener(s)
Prof. Chandan Singh Dalawat (Harish-Chandra Research Institute, Allahabad )
Dr. Supriya Pisolkar (Indian Institute of Science Education & Research, Pune )
Lecture 1 : Inverse and direct systems, limits and colimits, profinite groups, rings, and spaces.
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
Lecture 3 : Hensel's Lemma, squares in Z_p, raising to the exponent p, multiplicative representatives
The field of Q_p of p-adic numbers, squares and quadratic characters of Q_p^*, structure of the group Q_p^*.
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Lecture 5 : Hensel's Lemma, squares in Z_p, raising to the exponent p, multiplicative representatives
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The Hilbert symbol, Steinberg property, explicit formulae, the conic ax^2+by^2=1.
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Lecture 7 : Quadratic extensions of Q_p, their compositum, bijection with the set of quadratic characters of Q_p^*, Kummer theory, Artin-Schreier theory.
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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.
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Lecture 9 : Algebraic preliminaries, Galois theory, Artin-Schreier theory, reformulation of local quadratic reciprocity.
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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.
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Lecture 11 : p-fields, rings of integers, residue fields, uniformisers, Hensel's lemma, multiplicative representatives.
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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.
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Lecture 13 : Valuations, equivalence of valuations, rings of integers, residue fields, discrete valuations, uniformisers, ideals in the ring of integers, norms, equivalence of norms, completions.
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Lecture 14 : Hensel's lemma, extension of a complete discrete valuation to a finite extension field, integral closure of the ring of integers.
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Lecture 15 : Krasner's lemma, ramification index and residual degree, multiplicativity in towers.
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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.
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Lecture 17 : Eisenstein polynomials, totally ramified extensions, structure of tamely ramified extensions, parametrisation of tamely ramified extensions of a given degree.
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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.
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Lecture 19 : The trace map, the different and the discriminant, transitivity formulae, the different and the ramification, Serre's mass formula in the tame case.
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Lecture 20 : Inertia subgroup and ramification subgroup, higher ramification groups, lower numbering, ramification filtration on a subgroup.
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Lecture 21 : The ramification filtration on a quotient group, Hasse-Herbrandt functions, transitivity, roots on unity.
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Lecture 22 : Ramification filtration on cyclotomic extensions, ramification jumps, cyclic extensions, abelian extensions of prime exponent, Hasse-Arf.
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Lecture 23 : cyclic extensions of p-fields of characteristic 0, raising to the exponent p, filtration on the units modulo p-th powers, primitive roots of unity, the unramified F_p-line, ramification breaks of cyclic extensions.
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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.
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Lecture 25 : Parametrisation of solvable extensions of prime degree over an arbitrary field.
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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.
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Lecture 27 : The norm map and the ramification filtration.
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Lecture 28 : The local Kronecker-Weber theorem.
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Lecture 29 : Primitive solvable extensions of an arbitrary field.
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Lecture 30: Primitive solvable extensions of an arbitrary field.
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Lectures 31 : Irreducible F_p-representations of the absolute Galois group of a p-field.
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Lectures 32 : Irreducible F_p-representations of the absolute Galois group of a p-field.
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Lectures 33 : The Galois module structure of units modulo p-th powers in a tamely ramified split extension.
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Lectures 34 : The Galois module structure of units modulo p-th powers in a tamely ramified split extension.
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Lecture 35 : Primitive extensions of p-fields.
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Lecture 36 : Primitive extensions of p-fields.