In this part, you may assume any facts about the factorisation theory of Z[i], the ring of Gaussian Integers, and of Z provided that you state clearly the properties that you are using.
Let p be an integer prime for which there is an element a in Z with
. Write down a factorisation of p in Z[i] and show that it is a proper factorisation (that is, neither factor is a unit of Z[i]). Deduce that p is not a prime in the Gaussian integers. Show that the factors of p that you obtained are primes in Z[i].
Thoughts...
p = (a + i)(a - i)
I'm not certain about this but i think that a prime cant be a unit? So in Z that would be
?
If so then
hence since the only units in Z[i] are 1,-1,i ,-i. Neither factor is not a unit, i.e. proper.
p is not prime since p doesn't divide either factor. This is because
is imaginary and p is real.
Showing
are primes in Z[i]...
a + bi is prime if
is an integer prime of the form 4k+1. So how do i show
is of the form 4k+1?
Actually a thought has just occurred... Since
is prime,
has to be even, i.e. a is even. So let a be of the form 2n. Then we have
so
, hence
? So... ?
One final note...
could be even, but only if a=1, since 2 is the only even prime. But 1+i and 1-i are primes in Z[i] so that takes care of the a=odd case.