Page 41 - Advanced Linear Algebra

Preliminaries 25

Theorem 0.25 The integers form a principal ideal domain. In fact, any ideal ?
{
?
in is generated by the smallest positive integer a that is contained in .
Theorem 0.26 The ring -´%µ is a principal ideal domain. In fact, any ideal is
?
generated by the unique monic polynomial of smallest degree contained in .
?
Moreover, for polynomials ²%³Áà Á ²%³ ,

º ²%³Á à Á ²%³» ~ ºgcd ¸ ²%³Á à Á ²%³¹»

?
Proof. Let be an ideal in -´%µ and let ²%³ be a monic polynomial of
smallest degree in . First, we observe that there is only one such polynomial in
?
? ?. For if ²%³  is monic and deg ² ²%³³ ~ deg ² ²%³³ , then
²%³ ~ ²%³ c ²%³  ?
and since deg² ²%³³  deg² ²%³³ , we must have ²%³ ~ and so
²%³ ~ ²%³.
?
We show that ~ º ²%³» . Since ²%³  ? , we have º ²%³» ‹ ? . To establish
the reverse inclusion, if ²%³  ? , then dividing ²%³ by ²%³ gives
²%³ ~ ²%³ ²%³ b ²%³

where ²%³ ~ or  deg ²%³  deg ²%³ . But since is an ideal,
?
²%³ ~ ²%³ c ²%³ ²%³  ?
and so  deg ²%³  deg ²%³ is impossible. Hence, ²%³~ and

²%³ ~ ²%³ ²%³  º ²%³»
This shows that ‹ º ²%³» and so ~ º ²%³» .
?
?
To prove the second statement, let ? ~ º ²%³Áà Á ²%³» . Then, by what we

have just shown,
? ~ º ²%³Áà Á ²%³» ~ º ²%³»

where ²%³ is the unique monic polynomial ²%³ in of smallest degree. In
?
particular, since ²%³  º ²%³» , we have ²%³ “ ²%³ for each ~ Á à Á .

In other words, ²%³ is a common divisor of the ²%³ 's.

Moreover, if ²%³ “ ²%³ for all , then ²%³  º ²%³» for all , which implies

that
²%³  º ²%³» ~ º ²%³Á à Á ²%³» ‹ º ²%³»

and so ²%³ “ ²%³ . This shows that ²%³ is the greatest common divisor of the
²%³'s and completes the proof.