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Old 2019-05-07, 20:16   #1
bhelmes
 
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Default 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
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Old 2019-05-07, 20:41   #2
paulunderwood
 
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Quote:
Originally Posted by bhelmes View Post
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
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Old 2019-05-09, 10:34   #3
LaurV
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What's wrong with polar form?
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Old 2019-05-09, 11:53   #4
Dr Sardonicus
 
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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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