Eisenstein's Criterion is stated in Dummit and Foote as follows: (see attachment)
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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]
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The beginning of the proof reads as follows:
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... .,.. etc. etc.
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I will now take a specific example with R= Z as the integral domain concerned and P = (3) as the prime ideal in Z.
Also take (for example)
Now as the proof requires, reduce f(x) mod P
Now using D&F Proposition 2 (see attached) - namely we have
and so we to obtain we simply reduce the coefficients of f(x) by mod 3
Since
we have
The coset TEX] \overline{f(x)} [/TEX] would include elements such as and so on.
Can someone please confirm my working in this particular case of the Eisenstein proof is correct?
Peter