2 Attachment(s)

Specific example of Eisenstein's Theorem using R = Z

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