Specific example of Eisenstein's Theorem using R = Z

In summary, Eisenstein's Criterion states that if a polynomial in an integral domain satisfies certain conditions (such as having all coefficients in a prime ideal and the constant term not being in the square of that ideal), then it is irreducible. The proof for this criterion involves reducing the polynomial modulo the prime ideal and showing that it still satisfies the conditions. There may be confusion in understanding how the prime ideal applies to the polynomial in the proof, but the key point is that the prime ideal is being used to show that the reduced polynomial still satisfies the conditions for Eisenstein's Criterion.
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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

[TEX] f(x) = x^n + a_{n-1}x^{n-1} + ... ... + a_1x + a_0 [/TEX]

be a polynomial in R[x] (here [TEX] n \ge 1[/TEX] )

Suppose [TEX] a_{n-1}, ... ... a_1, a_0 [/TEX] are all elements of P and suppose [TEX] a_0 [/TEX] is not an element of [TEX] P^2 [/TEX].

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 [TEX] x^n = \overline{a(x)b(x)}[/TEX] 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) \(\displaystyle f(x) = x^3 + 9x^2 + 21x + 9 = (x+3) (x^2 +6x + 3) \)

Now as the proof requires, reduce f(x) mod P

Now using D&F Proposition 2 (see attached) - namely \(\displaystyle R[x]/(I) \cong R/I)[x] \) we have

\(\displaystyle Z[x]/(3) \cong (Z/(3))[x]\)

and so we to obtain \(\displaystyle \overline{f(x)} \) we simply reduce the coefficients of f(x) by mod 3

Since \(\displaystyle 9, 21 \in \overline{0} \)

we have \(\displaystyle \overline{f(x)} = \overline{x^3} \)

The coset \(\displaystyle \overline{f(x)} \) would include elements such as \(\displaystyle x^3 + 3, x^3 + 6x^2 + 24x - 3, ... ... \) and so on.

Can someone please confirm my working in this particular case of the Eisenstein proof is correct?

Peter

[This post is also on MHF]
 
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To discuss this example in terms of Eisenstein's Criterion, you need to point out that neither the hypothesis nor the conclusion of Eisenstein's theorem is satisfied here. The coefficients $9$ and $21$ are both multiples of $3$, but the constant term $9$ is a multiple of $3^2$ (contrary to Eisenstein's Criterion). And the polynomial $f(x)$ is not irreducible because it factorises as $ (x+3) (x^2 +6x + 3)$.
 
  • #3
Opalg said:
To discuss this example in terms of Eisenstein's Criterion, you need to point out that neither the hypothesis nor the conclusion of Eisenstein's theorem is satisfied here. The coefficients $9$ and $21$ are both multiples of $3$, but the constant term $9$ is a multiple of $3^2$ (contrary to Eisenstein's Criterion). And the polynomial $f(x)$ is not irreducible because it factorizes as $ (x+3) (x^2 +6x + 3)$.

Thanks Opalg.

Regarding my specific example, I think that my post was not completely clear in what I was attempting to demonstrate. I was taking a specific example and following the D&F proof on D&F page 310 - see attached - which assumes that f(x) is reducible and then proceeds to reduce f(x) modulo P. So, I took a reducible polynomial that (I thought) followed the Eisenstein rules for coefficients and then was focussed on showing how this led to the equation \(\displaystyle f(x) = \overline{a(x)b(x)} \) when f(x) is reduced modulo P.

Mind you as you point out I was wrong in allowing \(\displaystyle a_0 \in P^2 \). I am not sure it would really alter my exercise in establishing \(\displaystyle f(x) = \overline{a(x)b(x)} \)but I probably should have taken (say) \(\displaystyle f(x) = x^3 + 9x^2 + 24x + 18 = (x +3)(x^2 + 6x + 6) \) and then moved on (in parallel with or following the steps of D&F's proof to show that \(\displaystyle f(x) = \overline{a(x)b(x)} \) - since it is these steps that bother me.

*** Reflecting on the proof, I am confused by the following:

In D&F page 310 (see attached) we find the following:

"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 we obtain the equation \(\displaystyle f(x) = \overline{a(x)b(x)} \) in (R/P)[x] ... ... etc

My confusion is as follows:

P is a prime ideal in R (in my specific example P + (3) is a prime ideal in R = Z)

BUT!

P is not an ideal in R[x] --> so how can we reduce the equation f(x) = a(x)b(x) which is in R[x] by an ideal P which is not even in R[x]?

I would be extremely grateful if someone could clarify this situation for me

Peter
 

FAQ: Specific example of Eisenstein's Theorem using R = Z

What is Eisenstein's Theorem?

Eisenstein's Theorem is a mathematical theorem that states that a polynomial with integer coefficients can be factored into a product of polynomials of lower degree if and only if it is irreducible over the rational numbers.

How does Eisenstein's Theorem apply to R = Z?

In this specific example, R = Z refers to the ring of integers. This means that Eisenstein's Theorem can be used to determine whether a polynomial with integer coefficients can be factored into polynomials with integer coefficients.

Can you give an example of Eisenstein's Theorem using R = Z?

One example would be the polynomial x^3 - 6x^2 + 11x - 6. This polynomial is irreducible over the rational numbers, but using Eisenstein's Theorem with R = Z, we can factor it into (x-1)(x-2)(x-3), where each factor has integer coefficients.

What is the significance of Eisenstein's Theorem in mathematics?

Eisenstein's Theorem is significant because it provides a way to determine whether a polynomial with integer coefficients can be factored into simpler polynomials with integer coefficients. This has applications in number theory, algebraic geometry, and other areas of mathematics.

Are there any limitations to Eisenstein's Theorem?

One limitation of Eisenstein's Theorem is that it only applies to polynomials with integer coefficients. It cannot be used to factor polynomials with coefficients from other number systems, such as real or complex numbers. Additionally, it can only be used to determine irreducibility over the rational numbers, not over other fields.

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