Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics — algebraic geometry, in particular. This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Galois theory, Noetherian rings and modules, and rings of fractions. It covers the basics, starting with the divisibility theory in principal ideal domains and ending with the unit theorem, finiteness of the class number, and the more elementary theorems of Hilbert ramification theory. Numerous examples, applications, and exercises appear throughout the text.
Let A be an integral domain, K its field of fractions, x and y elements of K. We shall say that x divides y if there exists a â A such that y = ax. Equivalently, we say x is a divisor of y, y is a multiple of x; notation x | y. This relation between the elements of K depends in an essential manner on the ring A; if there is any confusion possible, we speak of divisibility in K with respect to A.
Given x â K we write Ax for the set of multiples of x. Thus we may write y â Ax in place of x | y, or Ay â Ax. The set Ax is called a principal fractional ideal of K with respect to A; if x â A, Ax is the (ordinary) principal ideal of A generated by x. As the relation of divisibility, x | y, is equivalent to the order relation Ay â Ax, divisibility possesses the following two properties associated with order relations.
On the other hand, if x | y and y | x, one cannot in general conclude that x = y; one has only that Ax = Ay, which (if y â 0) means that the quotient xyâ1 is an invertible element of A; in this case x and y are called associates; they are indistinguishable from the point of view of divisibility.
Example. The elements of K which are associates of 1 are the elements invertible in A; they are often called the units in A; they form a group under multiplication, and we shall denote this group A*. The determination of the units in a ring A is an interesting problem which we shall treat in the case where A is the ring of integers in a number field (see Chapter IV). Here are some simple examples:
(a)If A is a field, A* = A â (0).
(b)If A = Z, A* = {+1, â1}.
(c)The units in the ring of polynomials B = A[Xl, ..., Xn], A an integral domain, are the invertible constants; in other words B* = A*.
(d)The units in the ring of formal power series
are the formal power series whose constant term is invertible.
Definition 1. A ring A is called a principal ideal ring if it is an integral domain and if every ideal of A is principal.
We know that the ring Z of rational integers is a principal ideal ring. (Recall that any ideal
â (0) of Z contains a least integer b > 0. Dividing x â
by b and using the fact that Z is Euclidean, one sees that x is a multiple of b.) If K is a field we know that the ring K[X] of polynomials in one variable over K is a principal ideal ring (same method of proof: take a non-zero polynomial b(X) of lowest degree in the given ideal
â (0) and make use of the fact that K[X] is a Euclidean ring, i.e. the remainder under division of an arbitrary element of a by b(X) must be of lower degree than b(X) or zero, which implies zero). This general method shows that any âEuclidean ringâ (see [1], Chapter VI...
