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Example text

Furthermore, there exists an embedding cr of R in the algebraically closed field of all Puiseux series in fractional powers of liz, and this assists in enabling R to be presented explicitly. Each element ~ of R may be specified uniquely by its minimal polynomial ~ over k(z) and a finite number of terms in the Puiseux series to distinguish ~ from the other zeros further coefficients of the series of~. cr(~), cr(~), sufficient Since we may compute any and determine effectively whether a polynomial is irreducible over k(z), this description provides an explicit presentation of R as required.

For instance, the example in the next section would require the solution of some 157 equations in 132 unknowns using this second method, a most formidable task. We shall solve the equation using the first method, without recourse even to machine computation. 4 AN EXAMPLE In this section we shall determine all the solutions X, Y in k[Z] of the elliptic equation y2 X(X - z) (X - 2z + 1). This demonstration will provide a simple illustration of the general method of analYSis described in §§l, 3. It will be seen that an exactly Similar argument would apply to any equation of the form y2 G(X), where G has at least three zeros in k(z) with odd multiplicities; the number of steps involved in the calculation varies exponentially with H(G), that is, the degree of G in z.

Sllo-llo-b-a)= o. (ctSllo) = 1 is now automatically satisfied, since a lies in the same class as C and a:

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