# Polynomials in Zn

1. Oct 27, 2011

### Essnov

I am taking a first course in algebra and this is a problem in my textbook that has me stumped:

$$Fix \ a \in \mathbb{Z}_{n} \ and \ f \in \mathbb{Z}_{n}[x] \ with \ \deg(f) = m. Show \ there \ is \ h_{m-1} \in \mathbb{Z}_{p}[x] \ with \ \deg(h_{m-1}) \leq (m-1) \ so \ f = a_{m}(x - a)^m + h_{m-1}$$

We haven't covered polynomial rings or anything like that.

To be honest I don't understand how it makes sense to add coefficients in Zn with coefficients in Zp, although maybe it makes sense if p | n?

If it's true then:
$$f(a) = \sum_{k = 0}^{m} a_{k}a^{k} = h_{m-1}(a)$$
But it's unclear to me how that's possible.

Any hint at all is really appreciated.