Titlebook: Number Theory for Beginners; André Weil Textbook 1979 Springer-Verlag New York Inc. 1979 Zahlentheorie.algebra.form.mathematics.number the

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,§ VIII,Theorem II.1 shows that every subgroup . of . is either 0 or generated by its smallest element .0. in the latter case it is generated by . or also by —., but by no other element of . For cyclic groups, we have:

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,§ IX,In order to consider polynomials with coefficients in a field ., and equations over such fields, we begin by reviewing some elementary properties of polynomials over an arbitrary field .; these are independent of the nature of that field, and quite analogous to the properties of integers described above in §§ II, III, IV.

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,§ X,Let G be a group of order m. If, for every divisor d of m, there are no more than d elements of G satisfying x.=1, G is cyclic.

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Textbook 1979t the problem sessions so on became desultory. v vi Weekly notes were written up by Max Rosenlicht and issued week by week to the students. Rather than a literal reproduction of the course, they should be regarded as its skeleton; they were supplemented by references to stan- dard text-books on alge

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,§ XI,vial, we assume .≠0. If then, in the field ., . is a solution of x.=a, an element . of . is also a solution if and only if ...1. Therefore, if x.=a has a solution in ., it has as many solutions as . contains .. roots of unity, i.e. roots of ...1.

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,§ XII,onsisting of the classes (±1 mod . we apply to . and . the definitions and lemma of § VIII. If . is an element of ., it belongs to one and only one coset . this consists of the two elements (±. mod . there are . such cosets, viz., the cosets (±1 mod .), (±2 mod .),...(±. mod .). If, in each coset, w