Frequency of lectures, practice and lab: Recommended
Teaching Tools UniFi E-Learning: http://e-l.unifi.it
Office hours:
By appointment.
Department of Mathematics "Ulisse Dini"
Viale Morgagni, 67/a
50134 - Firenze (FI)
Tel: 055 4237108 Fax: 055 4237165
alessandro.scarselli@unifi.it, alessandro.scarselli@math.unifi.it
Type of Assessment
Oral
Course program
GROUPS WITH OPERATORS
Acceptable subgroups – Operatorial homomorphism and quotient groups – Homomorphism theorem – Chain and series – Zassenhaus’ lemma – Jordan-Hoelder’s theorem – Finity conditions – Commutator – Derivative series – Solvable group – Central chain – Nilpotent group – Direct product – Normal Endomorphism – Remak-Krull-Schmidt theorem – Semi-direct product.
FINITE GROUPS
p-groups and nilpotent groups – Schur’s and Zassenhaus’ theorem about breaking extensions – P. Hall’s theorem for solvable groups – Simple groups – An simplicity for n>4.
GALOIS’ THEOREM
Polynomial Galois’ group. Galois’ group as root substitution groups. An exercise about algebric closure of certain extension of perfect fields. Fundamental theorem of algebra. Radical extensions and solvability of their Galois’ group. Solvability of polynomial radicals. A quintic non solvable by radicals.
ABELIAN GROUPS AND MODULES
Irreducible modules – Modules completely reducible – Noetherian and artinian’s modules – Noetherian rings. Hilbert’s base theory. Modules on rings with principal ideals – Free modules with finite base – Invariance theorem of factors – Elemantary divisors – Finitely generated modules – Structure of finitely generated abelian’s groups.