Titlebook: Topics in Number Theory; J. S. Chahal Book 1988 Springer Science+Business Media New York 1988 Finite.Morphism.algebra.calculus.equation.fi

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Equations over Finite Fields,We have seen that for each prime ., there is a field F. of . elements. In fact, given any prime . and an integer . ≥ 1, there is one and only one field F. of . = .. elements. The field F. ⊇ F. and for each α in F., .α = 0. Conversely, any finite field is F., for some . = .. (cf. Ref. 18). The field F. is characterized by the property..

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Computation of the Mordell-Weil Group,.., .. is of finite order, ..,..., .. cannot be independent. For any elliptic curve . defined over ℚ the group .(ℚ) of rational points on . is finitely generated. The (.) ...(.) of . is defined to be the maximum number of independent elements in .(ℚ). In particular, ..(.) = 0 if and only if .(ℚ) is finite (consisting of points of finite order).

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The Mordell-Weil Theorem,e also gave a simpler proof, using the concepts he had introduced in his thesis, for the special case of elliptic curves. It is this proof that we shall be following (cf. Ref. 4 or 8). A very interesting account is in Cassels [1].