Gene Freudenburg

Actions of Ga on A3 defined by homogeneous derivations

The first example of an algebraic action of Ga on affine 3-space having maximal rank 3 is produced. Its fixed points consist of a single line in A3, and Ga is realized as an algebraic subgroup of Autk(A3) whose non-trivial elements are of degree 41. The corresponding derivation is homogeneous and irreducible of degree 4. Since triangulable actions are never of maximal rank, this action is non-triangulable. This action is embedded, for each n ≥ 3, into a Ga-action on An, in such a way that the resulting action has rank n, thus showing that algebraic Ga-actions on An having maximal rank exist for each n ≥ 3. n nAlso considered is the general case of a homogeneous locally nilpotent derivation on k[3]. The main tool here is the exponent of a polynomial relative to the derivation. By describing such derivations of type (2, d + 1), where d is the degree of the derivation, it is shown that actions induced by homogeneous derivations of degree less than four have rank at most 2. The rank 3 example mentioned above appears as a special case of Theorem 4.2.

Elements of Algebra. By John Stillwell

This book is a concise, self-contained introduction to abstract algebra which stresses its unifying role in geometry and number theory. There is a strong emphasis on historical motivation both to trace abstract concepts to their concrete roots, but also to show the power of new ideas to solve old problems. This approach shows algebra as an integral part of mathematics and makes this text more informative to both beginners and experts than others. Classical results of geometry and number theory (such as straightedge-and-compass construction and its relation to Fermat primes) are used to motivate and illustrate algebraic techniques, and classical algebra itself (solutions of cubic and quartic equations) is used to motivate the problem of solvability by radicals and its solution via Galois theory. Modern methods are used whenever they are clearer or more efficient, but technical machinery is introduced only when needed. The lively style and clear expositions make this book a pleasure to read and to learn from.