Quick question about ideals of field $F$ in $F[x]$ are principal.

Just got a question about ideals of $F[x]$ being principal with $F$ being a field.

The proof basically starts assuming $g(x) \in N \subset F[x]$ where $N$ is an ideal and $g(x)$ is a minimal degree element. The rest is just deducing the what the division algorithm says.

But my question is that shouldn't we let $g(x)$ be arbitrary? Why are we choosing the minimal element?

Solution 1:

You are trying to show the ideal $N$ is a principal ideal.

Let $g \in N$ have minimal degree.

The standard proof shows $N=(g)$, i.e., $N$ is the ideal generated by $g$.

If you choose $h \in N$ with $\deg(h)>\deg(g)$, you can form the principal ideal $(h)$, but it won't contain $g$, since any nonzero multiple of $h$ will have degree at least equal to $\deg(h)$.

Hence $N \ne (h)$.

The reason $g$ works is that when you use the division algorithm, the remainder, if nonzero, would be an element of $N$ with degree less than $\deg(g)$, which is impossible, since $\deg(g)$ is minimal (in $N$).

For the same reason $h$ fails, since when you use the division algorithm, the remainder, if nonzero, is an element of $N$ with degree less than $\deg(h)$, but $N$ has such elements (e.g., $g$), so there's no contradiction.

Solution 2:

The key idea behind the proof is as follows. In domains that enjoy division with remainder, ideals are closed under remainder, so by induction (Euclidean algorithm) they are also closed under gcd. Therefore the least (degree) element $g\in I$ necessarily divides every $f\in I$ since otherwise $\,\gcd(g,f)\in I$ is a smaller (degree) element in $I$, contra minimality of $g$.

Remark $ $ Presenting the descent as above by gcd (vs. mod or remainder) is not only clearer from a conceptual (divisibility) viewpoint, but it more easily leads the way to generalizations, e.g. for PIDs we can view ideal closure under gcds as reducing the number of prime factors of elements, yielding a gcd descent proof that nonzero ideals are generated by any nonzero member having the least number of prime factors.