Geometryof Numbers Proof of Goo tzky’s Four-Squares Theorem
نویسندگان
چکیده
The totally positive algebraic integers of certain number fields have been shown to be the sums of four squares of integers from their respective fields. The case ofQð ffiffiffi 5 p Þ was demonstrated by Götzky and the cases of Qð ffiffiffi 2 p Þ and Qð ffiffiffi 3 p Þ were demonstrated by Cohn. In the latter situation, only those integers with even coefficient on the radical term could possibly be represented by sums of squares. These results utilized modular functions in order to get the exact number of representations. Here a method of Grace is adapted to show the existence of a four-squares representation for Qð ffiffiffi 5 p Þ without, however, obtaining the number of these. Also, results about the representation of primes by sums of two squares are obtained for Qð ffiffiffi 5 p Þ: # 2002
منابع مشابه
An alternate proof of Cohn’s four squares theorem
While various techniques have been used to demonstrate the classical four squares theorem for the rational integers, the method of modular forms of two variables has been the standard way of dealing with sums of squares problems for integers in quadratic fields. The case of representations by sums of four squares in Qð ffiffiffi 5 p Þ was resolved by Götzky, while those of Qð ffiffiffi 2 p Þ an...
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