# Polynomials $f(x)$ of degree at most $5$ forming a ring and field

Show that the set of all polynomials $f(x)$ of degree at most $5$ with integer coefficients is a ring. Is the set of such polynomials a field?

I don't see how the ring of polynomials with degree at most $5$ is closed under multiplication. If I multiply $x^2$ and $x^5$ I do not get another polynomial of degree at most $5$.

• Well, they are not a ring, so it is not surprising that you cannot see it! – Mariano Suárez-Álvarez Apr 1 '11 at 5:16
• $\:\mathbb Z\:$ is the only subring of $\rm\:\mathbb Z[x]\:$ of bounded degree. Check the question. – Bill Dubuque Apr 1 '11 at 5:23

You're right, it's not a ring using addition and multiplication defined in the normal manner on $\mathbb{Z}[X]$. Are you sure that this is what the question asks? Does it perhaps introduce some equivalence relation, such as $x\equiv y$ iff $x-y\in (X^6)$?