We define an ideal $I$ to be the set of all polynomials with integer coefficients, with constant and linear terms divisible by $5$. (If $f(x) = a_0+a_1 x + \cdots \in I$, both $a_0$ and $a_1$ are divisible by $5$). I have already shown that $I$ is an ideal in $\mathbb Z[x]$. How do I go about finding polynomials that generate $I$?
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Imitate the method given here for the generators. Note that $2$ generators suffice in general, see this post. – Dietrich Burde Mar 16 '21 at 13:03
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Try to find a basis of this ideal as an abelian group and check if some elements of the basis can be linear combinations with coefficients in $\mathbb{Z}[x]$ of others. – Aphelli Mar 16 '21 at 13:03