Intended for graduate courses or for independent study, this book presents the basic theory of fields. The first part begins with a discussion of polynomials over a ring, the division algorithm, irreducibility, field extensions, and embeddings. The second part is devoted to Galo…
This paper is largely devoted to the behaviour of operators which are polynomials (in several variables) in full and one-sided shifts, together with their corresponding Toeplitz operators, with consequences for product type actions of compact connected Lie groups. Problems are o…
We study the representations of real homogeneous polynomials [italic]p([italic]x) = [italic]p([italic]x₁, ···, [italic]x[italic subscript]n) of even degree [italic]m ([italic]n-ary [italic]m-ics) as sums of [italic]m-th powers of linear forms and generalize the theory of positiv…
The 3x+1 problem, or Collatz problem, concerns the following seemingly innocent arithmetic procedure applied to integers: If an integer x is odd then "multiply by three and add one", while if it is even then "divide by two." The 3x+1 problem asks whether, starting from any posit…
