Groups and subgroups. Normal subgroups. Factor groups. Isomorphism theorems. Actions and permutation groups. Sylow theorems. Field extensions. Algebraic and trascendental extensions. Degree of an extension. Splitting fields, normal extensions and Galois extensions. Galois group. Galois correspondence. Finite fields.
Knowledge acquired:
Basics of group theory, concept of group action, elements of Galois’
theory, application of acquired knowledge for solving problems.
Skills acquired (at the end of the course):
The student will be able to solve basic problems in the theory of groups,
fields and equations
Prerequisites
Courses required: Algebra I
Courses recommended: Geometry I
Teaching Methods
CFU: 6
Total hours of the course (including the time spent in attending lectures,
seminars, private study, examinations, etc...): 150
Hours reserved to private study and other indivual formative activities:
84
Further information
Frequency of lectures, practice and lab: Recommended
Teaching Tools UniFi E-Learning: http://e-l.unifi.it
Type of Assessment
Written and oral.
Course program
Part I: Group Theory
Operations, powers, homomorphisms and isomorphisms.
Groups and subgroups. Cosets, subgroup's index and Lagrange's theorem. The symmetric group of degree 3. Cyclic groups. Order of elements. Normal subgroups and normality criteria. Quotient group. Kernel of an homomorphism. Homomorphism's theorems and Correspondence theorem. Products of subgroups. Direct products. Automorphisms. Conjugation. Dihedral groups. Permutation groups. Decomposition of a permutation into cycles. Transpositions and the alternating group. Cayley's theorem. Groups acting on sets. The "Orbit-stabilizer" theorem. The orbit's equation with applications (fixed points of p-groups). Conjugation action: the class formula and Sylow's theorems. Burnside's lemma.
Part II: Fields theory
Field's extentions. Algebraic and transcendental elements. Degree's formula. Algebraic extentions and algebraic closure. Splitting fields. Normal extetions. Multiple roots and separable extetions. Galois group, Galois extetions and order of their Galois group. Galois groups acting on polynomial's roots. Finite fields. Artin's Lemma and Galois connection.