d|z <=> d²|z²
Hey,
I have one problem, that's probably pretty easy to solve, but I'm stuck anyway
What I have to show is: d|z <=> d²|z²
My idea so far:
Suppose d|z and f|z².
Let z=p1^e1*...*pn^en, since the prime power decomposition of z is unique it follows d ϵ {p1^e1,....,pn^en}.
From z²=p1^2e1*....*pn^2en, it follows f ϵ {p1^2e1,....pn^2en}.
Does this already mean f=d², because the elements in the second set are the squares of the elements in the first set?
And how can I write it down properly?
Thanks in advance, Paul
Hey,
I have one problem, that's probably pretty easy to solve, but I'm stuck anyway
What I have to show is: d|z <=> d²|z²
My idea so far:
Suppose d|z and f|z².
Let z=p1^e1*...*pn^en, since the prime power decomposition of z is unique it follows d ϵ {p1^e1,....,pn^en}.
From z²=p1^2e1*....*pn^2en, it follows f ϵ {p1^2e1,....pn^2en}.
Does this already mean f=d², because the elements in the second set are the squares of the elements in the first set?
And how can I write it down properly?
Thanks in advance, Paul