| First Steps | p. 1 |
| Groups Acting on Vector Spaces and Coordinate Rings | p. 2 |
| V Versus V | p. 4 |
| Constructing Invariants | p. 6 |
| On Structures and Fundamental Questions | p. 7 |
| Bounds for Generating Sets | p. 7 |
| On the Structure of K[V)G: The Non-modular Case | p. 8 |
| Structure of K[V]G: Modular Case | p. 9 |
| Invariant Fraction Fields | p. 10 |
| Vector Invariants | p. 11 |
| Polarization and Restitution | p. 11 |
| The Role of the Cyclic Group Cp in Characteristic | p. 16 |
| Cp Represented on a 2 Dimensional Vector Space in Characteristic p | p. 17 |
| A Further Example: Cp Represented on 2 V2 in Characteristic p | p. 20 |
| The Vector Invariants of V2 | p. 23 |
| Elements of Algebraic Geometry and Commutative Algebra | p. 25 |
| The Zariski Topology | p. 25 |
| The Topological Space Spec(S) | p. 27 |
| Noetherian Rings | p. 27 |
| Localization and Fields of Fractions | p. 29 |
| Integral Extensions | p. 29 |
| Homogeneous Systems of Parameters | p. 30 |
| Regular Sequences | p. 31 |
| Cohen-Macaulay Rings | p. 32 |
| The Hilbert Series | p. 34 |
| Graded Nakayama Lemma | p. 35 |
| Hilbert Syzygy Theorem | p. 36 |
| Applications of Commutative Algebra to Invariant Theory | p. 39 |
| Homogeneous Systems of Parameters | p. 40 |
| Symmetric Functions | p. 44 |
| The Dickson Invariants | p. 45 |
| Upper Triangular Invariants | p. 46 |
| Noether's Bound | p. 46 |
| Representations of Modular Groups and Noether's Bound | p. 48 |
| Molien's Theorem | p. 50 |
| The Hilbert Series of the Regular Representation of the Klein Group | p. 51 |
| The Hilbert Series of the Regular Representation of C4 | p. 53 |
| Rings of Invariants of p-Groups Are Unique Factorization Domains | p. 54 |
| When the Fixed Point Subspace Is Large | p. 55 |
| Examples | p. 59 |
| The Cyclic Group of Order 2, the Regular Representation | p. 61 |
| A Diagonal Representation of C2 | p. 62 |
| Fraction Fields of Invariants of p-Groups | p. 62 |
| The Alternating Group | p. 64 |
| Invariants of Permutation Groups | p. 65 |
| Göbel's Theorem | p. 66 |
| The Ring of Invariants of the Regular Representation of the Klein Group | p. 69 |
| The Ring of Invariants of the Regular Representation of C4 | p. 72 |
| A 2 Dimensional Representation of C3, p = 2 | p. 75 |
| The Three Dimensional Modular Representation of Cp | p. 75 |
| Prior Knowledge of the Hilbert Series | p. 76 |
| Without the Use of the Hilbert Series | p. 78 |
| Monomial Orderings and Sagbi Bases | p. 83 |
| Sagbi Bases | p. 85 |
| Symmetric Polynomials | p. 89 |
| Finite Sagbi Bases | p. 91 |
| Sagbi Bases for Permutation Representations | p. 93 |
| Block Bases | p. 99 |
| A Block Basis for the Symmetric Group | p. 101 |
| Block Bases for p-Groups | p. 103 |
| The Cyclic Group Cp | p. 105 |
| Representations of Cp in Characteristic p | p. 105 |
| The Cp-Module Structure of Ñ[Vn] | p. 110 |
| Sharps and Flats | p. 110 |
| The Cp-Module Structure of Ñ[V] | p. 113 |
| The First Fundamental Theorem for V2 | p. 115 |
| Dyck Paths and Multi-Linear Invariants | p. 117 |
| Proof of Lemma 7.4.3 | p. 122 |
| Integral Invariants | p. 124 |
| Invariant Fraction Fields and Localized Invariants | p. 130 |
| Noether Number for Cp | p. 132 |
| Hilbert Functions | p. 138 |
| Polynomial Invariant Rings | p. 141 |
| Stong's Example | p. 147 |
| A Counterexample | p. 148 |
| Irreducible Modular Reflection Groups | p. 149 |
| Reflection Groups | p. 150 |
| Groups Generated by Homologies of Order Greater than 2 | p. 151 |
| Groups Generated by Transvections | p. 151 |
| The Transfer | p. 153 |
| The Transfer for Nakajima Groups | p. 164 |
| Cohen-Macaulay Invariant Rings of p-Groups | p. 170 |
| Differents in Modular Invariant Theory | p. 173 |
| Construction of the Various Different Ideals | p. 174 |
| Invariant Rings via Localization | p. 179 |
| Rings of Invariants which are Hypersurfaces | p. 185 |
| Separating Invariants | p. 191 |
| Relation Between K[V]G and Separating Subalgebras | p. 195 |
| Polynomial Separating Algebras and Serve's Theorem | p. 198 |
| Polarization and Separating Invariants | p. 201 |
| Using Sagbi Bases to Compute Rings of Invariants | p. 205 |
| Ladders | p. 211 |
| Group Cohomology | p. 213 |
| Cohomology and Invariant Theory | p. 214 |
| References | p. 223 |
| Index | p. 231 |
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