Suppose 32 = alpha*beta for alpha, beta reatively prime quadratic integers in Q[i] . Show that alpha = epsilon*gamma^2 for some unit epsilon and some quadratic integers gamma in Q[i].
Please help me . I got stuck with it!
I'm not an expert in this stuff, but I think you can use that factorization into primes is unique up to powers of the units, which are 1, -1, i and -i.
In this case you can use that:
2 = (1+i)*(1-i)
(1+i) and (1-i) are Gaussian prime numbers.
You can thus write 32 as:
32 = 2^5 = (1+i)^(5) * (1-i)^(5)
You want to split this into two factors that are squares:
32 = A^2 * B^2
This means that A and B must contain an even number of the prime factors (1+i) and (1-i):
A = (1+i)^(n) (1-i)^(m)
B = (1+i)^(r) (1-i)^(s)
A^2*B^2 = (1+i)^(2n+2r)*(1-i)^(2m+2s)
Now
32 = (1+i)^(5) * (1-i)^(5)
but we obviously need even powers of both primes! But factorization into primes is only unique up to factors of unit elements. (1-i) and (1+i) have the same norm and can be changed into each other by multiplication by units, e.g.
(1+i) = (1-i)*i
So, e.g.:
32 = (1+i)^(4) * (1-i)^(6)*i
Thank you very much for your help.
I will post the other question under the title " college math question" . Please do teach me. Thanks again.