In this thesis we study the essential dimension of the first Galois cohomology functors of finite groups. Following the result by N. A. Karpenko and A. S. Merkurjev about the essential dimension of finite p-groups over a field containing a primitive p-th root of unity, we compute the essential dimension of finite cyclic groups and finite abelian groups over a field containing all primitive p-th roots of unity for all prime divisors p of the order of the group. For the computation of the upper bound of the essential dimension we apply the techniques about affine group schemes, and for the lower bound we make use of the tools of canonical dimension and fibered categories. We also compute the essential dimension of small groups over the field of rational numbers to illustrate the behavior of essential dimension when the base field does not contain all relevant primitive roots of unity.

Interest in essential dimension problems has been growing over the past decade. In part, it is because the idea of essential dimension captures quite elegantly the problem of parametrizing a wide range of algebraic objects. But perhaps more, it is because the study of essential dimension requires most of the algebraic arsenal. What began as a problem in Galois cohomology and representation theory now has connections to versal torsors, stacks, motives, birational geometry, and invariant theory. This exposition will focus on just a small bit of this theory: algebraic tori and how they can be used to help us calculate the essential p-dimension for PGLn.

Let $G$ be an algebraic group over a field $F$ of arbitrary characteristic. A type $i$-degree $n$ Milnor $K$-invariant of $G$ is an element of $Inv^{i}(G, K^{M}_{n} \otimes_{\mathbb{Z}} \mathbb{Q}/\mathbb{Z})$. This goal of this thesis is to compute the group of degree $n$ Milnor $K$-invariants for certain algebraic groups $G$ when $n = 1, 2$. The main results are divided among two chapters. In Chapter 2, we compute the type-one (homomorphic) degree one Milnor $K$-invariants for algebraic groups of multiplicative type. We also compute the type-zero (homomorphic) degree one Milnor $K$-invariants for reductive groups. In Chapter 3, we compute the type-one degree two Milnor $K$-invariants for algebraic tori; this is joint work with Alexander Merkurjev.

Given a field F of arbitrary characteristic and an algebraic torus T/F we calculate degree 2 and 3 cohomological invariants of T with values in Q/Z(1) and Q_p/Z_p(2) respectively, the latter for p not equal to 2, char(F) and generalize the former to other algebraic groups. Moreover, we obtain descriptions of the corresponding unramified cohomology groups, and in particular of H³_nr(F(T), μ_n^{\otimes 2}) for n prime to 2 and char(F). In the process, we construct a useful short exact sequence for cohomological invariants and make connections with recent results on Chow groups of codimension 2.

For a finite group $G$, a $G$-module $M$, and a field $F$, an element $u\in H^d(G,M)$ is negligible over $F$ if for each field extension $L/F$ and every continuous group homomorphism from $\text{Gal}(L_{\text{sep}}/L)$ to $G$, $u$ is in the kernel of the induced homomorphism $H^d(G,M)\to H^d(L,M)$.We determine the group of negligible elements in $H^2(G, M)$ for every abelian group $M$ with trivial $G$-action in Chapter 3.

For $p$ a prime and a trivial $G$-action on the coefficients, the negligible elements in the cohomology ring $H^*(G,\mathbb{Z}/p\mathbb{Z})$ form an ideal.In Chapter 4, we show that when $p$ is odd or $p=2$ and either $|G|$ is odd or $F$ is not formally real, the Krull dimension of the quotient of mod $p$ cohomology by the negligible ideal is 0. However, when $p=2$, $|G|$ is even, and $F$ is formally real, the Krull dimension of the quotient of mod 2 cohomology of a finite 2-group by the negligible ideal is 1.

In Chapter 5, we compute generators of the negligible ideal in the mod $p$ cohomology of elementary abelian $p$-groups.We also partially compute generators of the negligible ideal in the mod $p$ cohomology of cyclic groups, finite abelian $p$-groups, dihedral groups, symmetric groups, and generalized quaternion groups under certain roots of unity assumptions.

This dissertation is concerned with calculating the group of degree three cohomological

invariants of a reductive group over a field of arbitrary characteristic. We prove a formula

for the group of degree three cohomological invariants of a split reductive group G with coefficients in Q/Z(2) over a field F of arbitrary characteristic. As an application, we then use this to define the group of reductive invariants of split semisimple groups, and compute

these groups in all (almost) simple cases. We additionally prove the existence of a discrete

relative motivic complex for any reductive group, which could be used to compute the degree

two and three invariants of arbitrary reductive groups.

This dissertation is concerned with calculating the group of unramified Brauer invariants of a finite group over a field of arbitrary characteristic. We present a formula for the group of degree-two cohomological invariants of a finite group G with coefficients in Q/Z(1) over a field F of arbitrary characteristic. We then specialize this formula to the case of decomposable invariants and compute the unramified decomposable cohomological invariants with coefficients in Q/Z(1) for various finite groups G. When the order of G is prime to the characteristic of F, we obtain a formula for the degree-two unramified normalized decomposable invariants of finite abelian G and certain nonabelian groups G. We additionally compute the degree-two unramified normalized decomposable invariants with coefficients in Q/Z(1) in the case that G is cyclic of order p over a field of characteristic p.