Noncommutative rational invariantsPodlogar, Gregor (Avtor) Klep, Igor (Mentor) Noncommutative rational invariantsnoncommutative Noether’s problemautomorphisms of free skew-fieldRational functions in $d$ variables over a field ${\mathbb F}$ are actual (partial) functions from ${\mathbb F}^d$ to ${\mathbb F}$ that can be formed using coordinate functions and rational operations (addition, scalar multiplication, multiplication, inversion). Such functions form a field. Noncommutative rational functions in $d$ variables over ${\mathbb F}$ are partial functions from $d$-tuples of equally sized square matrices over ${\mathbb F}$ to matrices of the same size that can be formed using coordinate functions and rational operations. Such functions form a skew-field where every relation between the variables follows from the existence of inverses of nonzero elements, hence, the skew-field of noncommutative rational functions is also called a free skew-field. One of the major problems of invariant theory is Noether’s problem – given an action of a finite group on a field of rational functions, is the field of invariant functions isomorphic to a field of rational functions? In the thesis, we investigate a noncommutative version of Noether’s problem – given an action of a finite group on a free skew-field, is the skew-field of invariant functions free, i.e., isomorphic to a free skew-field? We study the actions of finite abelian groups on the free skew-field over ${\mathbb C}$ and ${\mathbb R}$ that are given by linear representations and show that their invariant skew-subfields are always free. We define complete representations – a type of linear representation of solvable groups that admit an inductive extension of the result for linear actions of abelian groups. For example, the standard representations of the symmetric groups $S_3$ and $S_4$ are complete. We also investigate so-called multiplicative actions of finite cyclic groups – actions that are defined by an automorphism of a free group and show that they are equivalent to linear actions. We give some interesting examples of invariants of cyclic groups over ${\mathbb Q}$ and invariants of the general linear group. The last part of the thesis is more group theoretic. We give an alternative characterisation of the groups that admit complete representations and name them totally pseudo-unramified groups. We present some group theoretic properties of totally pseudo-unramified groups and classify totally pseudo-unramified $p$-groups of rank up to five.20232023-04-23 08:15:13Doktorsko delo/naloga145588sl