On lower degree bounds for vector invariants over finite fields

The purpose of this thesis is to obtain a lower degree bound in modular invariant theory for a special case. More precisely, let G be any group and k be a finite field of positive characteristic p such that p divides |G| . We prove that if an invariant which has degree at most p —1 with respect to each variable can be written as a polynomial in orbit sums of monomials, then the invariant ring of m copies of the vector space V over k with dimV = n requires a generator of degree ^ ^ ^ provided that m > n where t and rii depends on the representation of G such that |'^'| < t < n + l and 2 < ni < p.