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I know this sounds like a basic question but I'm really confused. What does the notation $\langle x,3\rangle$ refer to for $\mathbb Z[x]$? Can someone write out what this is?

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    It's a mystery.2011-05-13

2 Answers 2

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The notation means the ideal of $\mathbb{Z}[x]$ generated by the elements $3$ and $x$ (see here) $\langle x,3\rangle=\{xf+3g\mid f,g\in\mathbb{Z}[x]\}=\{a_nx^n+\cdots+a_1x+a_0\mid a_i\in\mathbb{Z}\text{ and }3\text{ divides }a_0\}.$

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    @Gerry, the OP did ask for the fish...2011-05-13
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$\rm \langle x,3\rangle\:$ could mean a few things. Here are some possibilities, listed most-likely first. It could denote the ideal $\rm\: x\ \mathbb Z[x] + 3\ \mathbb Z[x]\:.\:$ Or it could denote the additive subgroup $\rm\:x\ \mathbb Z + 3\ \mathbb Z\:.\:$ Finally it could denote $\rm\:gcd(x,3)\:,\:$ though the gcd is more commonly denoted as $\rm\:(x,3)\:$ or $\rm\:[x,3]\:.$