Hilbert's Theorem 90 - Examples

Examples

Let L/K be the quadratic extension . The Galois group is cyclic of order 2, its generator s is acting via conjugation:

An element in L has norm . An element of norm one corresponds to a rational solution of the equation a2 +b2=1 or in other words, a point with rational coordinates on the unit circle. Hilbert's Theorem 90 then states that every element y of norm one can be parametrized (with integral c,d) as

y={{c+di}\over{c-di}}={{c^2-d^2}\over{c^2+d^2}}+{2dc\over{c^2+d^2}}i

which may be viewed as a rational parametrization of the rational points on the unit circle. Rational points on the unit circle correspond to Pythagorean triples, i.e. triples of integers satisfying .

Read more about this topic:  Hilbert's Theorem 90

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