By P. M. Cohn FRS (auth.)

ISBN-10: 0412361906

ISBN-13: 9780412361906

ISBN-10: 1489934448

ISBN-13: 9781489934444

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**Additional info for Algebraic Numbers and Algebraic Functions**

**Example text**

Clearly B is again a valuation ring; let m, n be the maximal ideals of A, B respectively. Then p =A n n is a non-zero prime ideal in A and if u e B\A, then u- I e m\n hence lJ c m. This contradicts (b), so A must be maximal. (c) => (a). Assume that Ais as in (c) and let ce A, c - I E A. Then A [c- I ] is a ring strict1y containing A, hence A [c- I] = K. Now K cannot be finitely generated as an A-module, for if it were, then every element of K would have the form ac-rn for aeA and a fixed m. Thus c- rn - I = ac- rn and hence c- I = ae A, a contradiction.

Since m'A ~ na, we have mn''A ~ nn'a, hence m'n'A ~ mn''A ~ nn'a and therefore m'/n' = m'n/n'n E L(a). This shows L(a) to be a lower segment in Q; it follows that its complement U(a) is an upper segment. We thus have a Dedekind cut. Let us define <\l(a) as the corresponding real number, thus L(a) consists of the rational numbers ~ <\l(a) and U(a) of the rational numbers > <\l(a). We claim that a ~ <\l(a) is an order-embedding of r in R. Given a, a' E r, if m/n E L(a), m'/n' E L(a'), then m'A ~ na, m''A ~ n'a', hence mn''A ~ nn'a, m' n'A ~ nn' a' and so (mn' + m'n) 'A ~ nn'(a + a'), E L(a + a').

An abelian group which has a total ordering which is preserved by the group operation. We shall usually write r additively and call an element a of r positive if a> O. A subset L of r is said to be convex if a, ~E L, say a ,,;:; ~, and a ,,;:; A ,,;:; ~ for some A E r implies AE L. In particular, we shall be concerned with convex subgroups (sometimes called 'isolated' subgroups). We note that the whole group and the trivial subgroup {O} are always convex. 1 below. By an orderhomomorphism of ordered groups we understand a homomorphism respecting the order; order-endomorphism, order-embedding and order-automorphism are defined similarly.

### Algebraic Numbers and Algebraic Functions by P. M. Cohn FRS (auth.)

by Ronald

4.4