By Andre Weil
From the reports: "L.R. Shafarevich confirmed me the 1st variation […] and stated that this ebook could be any further the ebook approximately category box concept. actually it's through a ways the main whole remedy of the most theorems of algebraic quantity idea, together with functionality fields over finite consistent fields, that seemed in publication form." Zentralblatt MATH
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COROLLARY 3. Let f be any linear form on V, and N a K-norm on V. Then there is v, f0 in V, such that N(v)- ’ mod,(f (v)) G NV,)(2) for all v#O in I/: ’ moddf (vl)) This is a special case of corollary 2, that corollary being applied to the left-hand side of (2). If one denotes by N*(f) the right-hand side of (2), then N*(f) is the smallest positive number such that mh(f (v)) d N*(f). e. on the right vector-space made up of the linear forms on V (where the addition is the obvious one, and the scalar multiplication is defined by putting (fa)(v)=f(v)a when f is such a form, and aEK).
V,} ; put L= x R Ui. IJIm; therefore ‘3JI is isomorphic to L/M, where M is the kernel of that morphism. Apply now corollary 1 to L and M; as M c L, we have vj > 0 for 1
Moreover, the above calculations show also that this expression is unique, which proves our assertions. 5 4. Lattices over R. The concept of lattice, as developed for p-fields in $0 1-2, cannot be applied to R-fields. The appropriate concept is here as follows : DEFINITION 3. By an R-lattice in a vector-space V of finite dimension over an R-field, we understand a discrete subgroup L of V such that V/L is compact. We have to recall here some elementary facts about discrete subgroups. Let G be a topological group, r a discrete subgroup of G, and cp the canonical mapping of G onto G/T.
Basic Number Theory by Andre Weil