By Abraham Robinson (Eds.)
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Additional resources for Complete Theories
24, c is different from all elements of F . Moreover, since F is algebraically closed, c must be algebraically independent of F. It follows that F* includes a subfield F" which includes c and which is isomorphic to F' under an isomorphism which centralises the elements of P as well as the constant c . The fields obtained by adjoining c to F in F' and F" though certainly isomorphic, do not necessarily coincide a priori, but we may without loss of generality identify corresponding elements. Accordingly, we shall speak of the field F ( c ) as being included in both F' and F".
18. 16. 19) is provedin ref. 19. Let F be a field with valuation in a group G and let F' be a finite normal algebraic extension of F . Let GI and Gi be two extensions of G in each of which there is given a valuation of F' which is a continuation of the valuation of F. Then there exists an automorphism of F' which centralises the elements of F and an order preserving isomorphism between GI and Gi which centralises the elements of G such that corresponding elements of F' are valued by corresponding elements of GY and G,'.
29 yields v(y) = v(z-u,) - v(uo)= b - b = 0. Also, putting ci=(a,-ul)/uo for i = 2 , ... 29, v(y-c4) =v = i = 2 , ... m. 31. , m. 31 is satisfied in M' by y = (c-u,)/uo, and we have to solve it in M . 29, x = u a + % . e. e. the set of all r E F for which v(r)>O. Then the quotient ring FR=R/J is known to be a field. , ym the elements of FR which contain c2, .. , c, respectively. e. , m since w(ci)= 0. 32. In order to prove that this system possesses a solution in FR, it is sufficient to show that FR is infinite.