Creating a Field from a Principal Ideal Domain

Theorem

• Let $(R,+,\times )$ be a PID
• Let $a\in \mathbb{R}$ be a prime element
• Then, $\mathbb{R}\mathrm{mod}a$ forms a field

Proof

• Let $a\in \mathbb{R}$
• Let $x=qa+r$
• Then, $x\mathrm{mod}a=r$
• As $(R,+,\times )$ is a principal ideal domain
• We must show that $\forall \in R\mathrm{mod}a$, s.t $d\ne 0,\exists b\in R\mathrm{mod}a$ s.t $d,b=1$