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I am looking at https://math.berkeley.edu/~ecarter/Summer08/110/notes/lec19.pdf

and trying understand the proof for the theorem that states:

Given a nonzero ideal I in $P(F)$, there is a monic polynomial $p(t)$ such that

$$I=\left\{q(t)p(t) | q(t) \in P(F) \right\}$$

Proof goes like

Since $I$ contains at least one nonzero element, we can let $p(t)$ be a nonzero element of $I$ of minimum degree. We can show that every element of $I$ is a multiple of $p(t)$. Let $f(t) \in I$. Then there exist $q(t),r(t) \in P(F)$ such that

$$f(t) = q(t)p(t) + r(t)$$ and deg $r(t)$ < deg $p(t)$. Since $p(t) \in I$, so is $q(t)p(t)$, and therefore so is $$r(t) = f(t) -q(t)p(t)$$.

By the choice of $p(t),r(t) = 0$, so $f(t)$ is a multiple of $p(t)$.

What I am having trouble understanding is why $r(t)$ must equal $0$. If someone can explain why it is, I'd appreciate it.

Jeff
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1 Answers1

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The key idea is that ideals are closed under remainder (mod), so the least (degree) $\,p\in I\,$ must divide every $\,f\in I,\,$ else $\,0\neq f\ {\rm mod}\ p\,$ is in $\,I\,$ and smaller than $\,p,\,$ contra minimality of $\,p.\,$

This is essentially the same proof as in $\,\Bbb Z.\,$ In any domain enjoying a division algorithm with smaller remainder, it shows that all ideals $\ne 0$ are principal, generated by an element of least size.

Such domains are called Euclidean domains since, like in $\,\Bbb Z,\,$ the division algorithm yields a Euclidean algorithm to compute gcds. The descent in the above proof can be interpreted constructively as computing a generator of $\,I\,$ by computing the gcd of its elements (by taking repeated remainders).

Remark $\ $ The idea extends to PIDs: (Dedekind-Hasse criterion) a domain $\rm\,D\,$ is a PID iff given $\rm\:0\neq a, b \in D,\:$ either $\rm\:a\:|\:b\:$ or some D-linear combination $\rm\:a\,d+b\,c\:$ is "smaller" than $\rm\,a.\,$

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Bill Dubuque
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