polynomials

**ricky** · February 10th 2011, 02:19 PM

how do I show that the set of polynomials with interger coefficients is countable?

**DrSteve** · February 10th 2011, 02:45 PM

See my response to your similar question about the algebraic numbers. If you're still stuck, say where you're stuck and I'll help you with the details.

**Drexel28** · February 10th 2011, 10:16 PM

Originally Posted by ricky

how do I show that the set of polynomials with interger coefficients is countable?

The set of all such polynomials of degree less than or equal to $n$ , denoted by $\mathbb{Z}_n[x]$ is clearly countable via the obvious identification $\displaystyle f:\mathbb{Z}_n[x]\to\mathbb{Z}^{n+1}:\sum_{k=0}^{n}\alpha_k x^k\mapsto (\alpha_0,\cdots,\alpha_k)$ . Note then that $\displaystyle \mathbb{Z}[x]=\bigcup_{n\in\mathbb{N}}\mathbb{Z}_n[x]$ and so..

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