Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2012 / 3 / b

A Euclidean domain is an integral domain $R$ with a function $d:R\setminus \{0\}\to {\mathbb{N}}_0$ such that, for $a\in R$ and nonzero $b\in R$, there exist $q,r\in R$ with $a=qb+r$ and either $r=0$ or $d(r)<d(b)$. This is the Euclidean-ring convention used here.

If an ideal $I$ is zero, it is generated by zero. Otherwise choose nonzero $a\in I$ with $d(a)$ minimal. For any $b\in I$, Euclidean division gives $b=qa+r$, with $r\in I$. A nonzero remainder would have $d(r)<d(a)$, contradicting the choice. Hence $r=0$ and $b\in aR$. The reverse inclusion follows from $a\in I$, provingNow suppose $a$ is an irreducible element. Its principal ideal is proper because $a$ is not a unit. If $aR\subseteq J\subseteq R$, write $J=bR$ by the result just proved. Then $a=bc$ for some $c\in R$. Irreducibility gives either $b$ a unit, so $J=R$, or $c$ a unit, so $bR=aR$. There is no intermediate proper ideal, and therefore $aR$ is a maximal ideal.

For a field $F$, the polynomial ring $F[X]$ is an integral domain and the polynomial division algorithm gives $A=QB+R$ with $R=0$ or $\mathrm{deg}R<\mathrm{deg}B$. Thus the polynomial degree is a Euclidean function, proving $F[X]$ is a Euclidean domain.