Unique factorization in quadratic domains
Namboodiri, M. S. T.
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Among quadratic domains some have unique factorization property and others don't. Gauss' conjecture that there are infinitely many real quadratic fields and only nine imaginary quadratic fields having unique factorization property, has not yet been proved. But there has been success in finding almost all complex quadratic domains having this property. The purpose of this thesis is to discuss unique factorization property of quadratic domains in general, with particular reference to complex quadratic domains. The basic theorem which is proved here is that a quadratic domain is a unique factorization domain if and only if it is a principal ideal domain. Euclidean domain is defined and some Euclidean and non-Euclidean domains are given for illustration. Being a Euclidean domain is only a sufficient but not a necessary condition for unique factorization. By making use of the definition of a Euclidean domain all complex quadratic Euclidean domains are found. Equivalent ideals, ideal classes and class number are also defined. The class number of a field is unity if and only if the domain is a principal ideal domain; i.e., if and only if it is a unique factorization domain. This fact is the key to our discussion here; the fact that the size of the class number gives a measure of how far is our domain from the factorization domain. To determine the class number of a given domain Minkowski's theorem is used. This theorem establishes the existence of an integral ideal in every class such that the norm of the ideal is always less than the absolute value of the square root of the discriminant of the field. Finally a necessary and sufficient condition for a complex qudratic field to have class number unity, is provided by the theorem on Euler's polynomial. In view of the findings of Lehmer, Heilbronn and Linfoot, it is concluded that there are nine complex quadratic fields of class number 1. If at all there is any more, there is only one more d less than zero for which Q (√d)has class number 1.
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