91Ó°ÊÓ

Let \(a\) and \(b\) be nonzero integers. We can find nonzero integers \(q\) and \(r\) such that \(a=q b+r\), where \(0 \leq r

Prove that \(\sqrt[n]{m}\) is irrational if \(m\) is not the \(n\)th power of an integer.

Define the least common multiple of two integers \(a\) and \(b\) to be an integer \(m\) such that \(a|m, b| m\), and \(m\) divides every common multiple of \(a\) and \(b\). Show that such an \(m\) exists. It is determined up to sign. We shall denote it by \([a, b]\).

Show that in any integral domain a prime element is irreducible.