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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 $5$.

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  • $\begingroup$ Well, they are not a ring, so it is not surprising that you cannot see it! $\endgroup$ – Mariano Suárez-Álvarez Apr 1 '11 at 5:16
  • $\begingroup$ $\:\mathbb Z\:$ is the only subring of $\rm\:\mathbb Z[x]\:$ of bounded degree. Check the question. $\endgroup$ – Bill Dubuque Apr 1 '11 at 5:23
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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 $x-y\in (X^6)$?

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  • $\begingroup$ The question was a typo on my professor's part. He wanted to show that Z[X]/x^6=1 was a ring. $\endgroup$ – user2888 Apr 1 '11 at 19:44
  • $\begingroup$ Yeah, that's what I figured. $\endgroup$ – Alex Becker Apr 2 '11 at 3:08

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