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)
Proposition 13 (Eisenstein's Criterion) Let P be a prime ideal of the integral domain R and let
be a polynomial in R[x] (here )
Suppose are all elements of P and suppose is not an element of .
Then f(x) is irreducible in R[x]
The proof begins as follows: (see attachment)
Proof: Suppose f(x) were reducible, say f(x) = a(x)b(x) in R[x] where a(x) and b(x) are nonconstant polynomials.
Reducing the equation modulo P and using the assumptions on the coefficients of f(x) we obtain the equation 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 and 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 of f(x) as the product of these two would be an element of , 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 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 and 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