Let k be a field and let p, q ∈ N be two prime numbers such that p · 1 = q · 1 = 0. Show that p = q.

Question

My current train of though is letting p =/= q then proving that q must be divisible by p, the contradiction then being that q is prime. But I'm not sure how to go about doing this.

Answer 1

If $p \neq q$ then we have $x,y$ such that $xp+yq=1$ since $\mathbb{Z}$ is a Euclidean domain. But then $1 = (xp+yq)1 = 0$, a contradiction.

Answer 2

If you assume $p \neq q$, and let $p > q,$ then $(p-q)\cdot 1 \neq 0$ since $p-q$ is not prime.

But $(p-q)\cdot 1 = p \cdot 1 - q \cdot 1 = 0,$ a contradiction.