20190507, 20:16  #1 
Mar 2016
1A0_{16} 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)(2i) alpha = arc tan (2/1) Would be nice to get a link or a clear answer, Greetings from the gaussian primes Bernhard 
20190507, 20:41  #2  
Sep 2002
Database er0rr
1119_{16} Posts 
Quote:


20190509, 10:34  #3 
Romulan Interpreter
"name field"
Jun 2011
Thailand
24005_{8} Posts 
What's wrong with polar form?

20190509, 11:53  #4 
Feb 2017
Nowhere
13722_{8} 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. (P^{k} is not a rational integer for any integer k other than 0.)
However, it is also easily shown that, if p_{1}, p_{2}, ..., p_{k} are k distinct prime numbers congruent to 1 (mod 4), P_{j} = (a_{j} + i*b_{j})R is a prime divisor of p_{j}R, j = 1 to k, and x_{j} = arg(a_{j} + i*b_{j})/pi < the circle number, then the x_{j} are Zlinearly independent  a much stronger result. This result follows from unique factorization in R  the product of integer powers of the P_{j} 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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