I am reading Dummit and Foote Section 9.4 Irreducibility Criteria. In particular I am struggling to follow the proof of Eisenstein's Criteria (pages 309-310 - see attached).

Eisenstein's Criterion is stated in Dummit and Foote as follows: (see attachment)

Let P be a prime ideal of the integral domain R and letProposition 13 (Eisenstein's Criterion)

$\displaystyle f(x) = x^n + a_{n-1}x^{n-1} + ... ... + a_1x + a_0 $

be a polynomial in R[x] (here $\displaystyle n \ge 1$ )

Suppose $\displaystyle a_{n-1}, ... ... a_1, a_0 $ are all elements of P and suppose $\displaystyle a_0 $ is not an element of $\displaystyle P^2 $.

Then f(x) is irreducible in R[x]

The proof begins as follows: (see attachment)

Suppose f(x) were reducible, say f(x) = a(x)b(x) in R[x] where a(x) and b(x) are nonconstant polynomials.Proof:

Reducing the equation modulo P and using the assumptions on the coefficients of f(x) we obtain the equation $\displaystyle x^n = \overline{a(x)b(x)}$ in (R/P)[x] where the bar denotes polynomials with coefficients reduced modulo P. Since P is a prime ideal, R/P is an integral domain, and it follows that $\displaystyle \overline{a(x)} $ and $\displaystyle \overline{b(x)} $ have zero constant term i.e. the constant terms of both a(x) and b(x) are elements of P. But then the constant term $\displaystyle a_0 $ of f(x) as the product of these two would be an element of $\displaystyle P^2 $, a contradiction.

Questions/Issues

(1) I cannot see exactly how the following follows:

"Reducing the equation modulo P and using the assumptions on the coefficients of f(x) we obtain the equation $\displaystyle x^n = \overline{a(x)b(x)}$ in (R/P)[x]"

I am struggling to see just exactly how this follows: can someone please clarify this?

(2) I am somewhat unsure of the following:

"t follows that $\displaystyle \overline{a(x)} $ and $\displaystyle \overline{b(x)} $ have zero constant term"

This probably is a consequence of the mechanics of the reduction modulo P process but can someone please clarify exactly how it follows?

Peter