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 2019-05-07, 20:16 #1 bhelmes     Mar 2016 1A016 Posts angle for gaussian primes ? A peaceful and pleasant evening, is it possible to calculate an angle for gaussian primes ? for example 5=(2+i)(2-i) alpha = arc tan (2/1) Would be nice to get a link or a clear answer, Greetings from the gaussian primes Bernhard
2019-05-07, 20:41   #2
paulunderwood

Sep 2002
Database er0rr

111916 Posts

Quote:
 Originally Posted by bhelmes A peaceful and pleasant evening, is it possible to calculate an angle for gaussian primes ? for example 5=(2+i)(2-i) alpha = arc tan (2/1) Would be nice to get a link or a clear answer, Greetings from the gaussian primes Bernhard
See: http://mathworld.wolfram.com/ArgandDiagram.html

 2019-05-09, 10:34 #3 LaurV Romulan Interpreter     "name field" Jun 2011 Thailand 240058 Posts What's wrong with polar form?
 2019-05-09, 11:53 #4 Dr Sardonicus     Feb 2017 Nowhere 137228 Posts If R = Z[i], p is a prime number, p == 1 (mod 4), then pR = PP', the product of two conjugate prime ideals. If P = (a + b*i)R, it is easily shown that the argument of a + b*i is not a rational multiple of the number pi. (Pk is not a rational integer for any integer k other than 0.) However, it is also easily shown that, if p1, p2, ..., pk are k distinct prime numbers congruent to 1 (mod 4), Pj = (aj + i*bj)R is a prime divisor of pjR, j = 1 to k, and xj = arg(aj + i*bj)/pi <-- the circle number, then the xj are Z-linearly independent -- a much stronger result. This result follows from unique factorization in R -- the product of integer powers of the Pj is not a rational number unless all the exponents are 0. Why the above argument does not apply to the prime divisor of 2R is left as an exercise for the reader.

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