In this course, we will cover the basics of what is called algebraic number theory. Just as number theory is often described as the study of the integers, algebraic number theory may be loosely described as the study of certain subrings of fields K with [K : Q] < ∞; these rings tend to act as natural generalizations of the integers. However, although algebraic number theory has evolved into a s...

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show first-order undecidability. In particular, we show that the following propositions hold. (1) For any rational prime q and any positive rational integer m, algebraic integers are definable in any Galois extension of ...