19/10/27 12:34:53.72 EUeYkluT.net
>>266
つづき
An example
The following example illustrates the notions introduced above. In order to compute the ramification index of Q(x), where
f(x) = x^3 - x - 1 = 0,
at 23, it suffices to consider the field extension Q23(x) / Q23. Up to 529 = 232 (i.e., modulo 529) f can be factored as
f(x) = (x + 181)(x^2 - 181x - 38) = gh.
Substituting x = y + 10 in the first factor g modulo 529 yields y + 191, so the valuation |?y?|g for y given by g is |?-191?|23 = 1. On the other hand, the same substitution in h yields y2 - 161y - 161 modulo 529. Since 161 = 7?×?23,
|y|h = √?161?23 = 1 / √23.
Since possible values for the absolute value of the place defined by the factor h are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the ramification index of the field extension at 23 is two.
The valuations of any element of F can be computed in this way using resultants. If, for example y = x^2 - x - 1, using the resultant to eliminate x between this relationship and f = x^3 - x - 1 = 0 gives y^3 - 5y^2 + 4y - 1 = 0.
If instead we eliminate with respect to the factors g and h of f, we obtain the corresponding factors for the polynomial for y, and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for g and h (which are both 1 in this instance.)
つづく