For Lucas-style tests on a number " N ", we work in the multiplicative group of a quadratic extension of the integers modulo " N "; if " N " is prime, the order of this multiplicative group is " N " 2 " 1, it has a subgroup of order " N " + 1, and we try to find a generator for that subgroup.
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Similarly to the case of algebraic closure, there is an analogous theorem for real closure : if " K " is a real closed field, then the field of Puiseux series over " K " is the real closure of the field of formal Laurent series over " K " . ( This implies the former theorem since any algebraically closed field of characteristic zero is the unique quadratic extension of some real-closed field .)
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