By P. M. Cohn FRS (auth.)
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Extra info for Algebraic Numbers and Algebraic Functions
Let K be a field with a valuation v and residue dass field k. Show that if k is archimedean ordered, then K has an ordering inducing the valuation v (as in Exercise 4). Obtain a generalization to the case where the residue dass field is an arbitrary ordered field. 6. Let K be a field with a valuation ring V. If P is a minimal non-zero prime ideal of V, show that the localization Vp is a valuation corresponding to a rank 1 valuation of K. 2 Extensions A central part of our topic is the study of extensions of rings of algebraic integers and of algebraic functions.
Show that any analytie homomorphism from A to a eomplete absolute valued ring S ean be extended in just one way to an analytic homomorphism of R into S. 6. 2. 4 VALUATIONS, VALUATION RINGS AND PLACES. 3 we obtained a fairly eomplete pieture of fields with an arehimedean absolute value. We now turn to eonsider the non-archimedean case in more detail. 2) is no Ion ger needed. This means that we can take the values to lie in any ordered multiplieative group, but for the moment we shall eonfine ourselves to the ease of the real numbers.
An, ao -j:. 0; in terms of y we have f= aoy-n + ... + an = y- n(ao + alY + ... + an/). Here the second factor is a unit and y is a uniformizer, therefore V (f) = - n = - degf, where deg refers to the degree in x. Thus for = f/g we have v (
Algebraic Numbers and Algebraic Functions by P. M. Cohn FRS (auth.)