MHB Field Extensions, Polynomial Rings and Eisenstein's Criterion

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Eisenstein's Criterion applies to polynomials over an integral domain, and since the subring of integers, ℤ, is an integral domain, the polynomial p(x) = x² - 2 can be evaluated using this criterion despite being considered over the field ℚ. The example provided in Dummit and Foote demonstrates that while ℚ is a field, it contains the integral domain ℤ, allowing the application of Eisenstein's Criterion. Additionally, any field is inherently an integral domain, which supports the validity of using the criterion in this context. Therefore, the confusion arises from the distinction between the field and its integral domain subring. Understanding this relationship clarifies how Eisenstein's Criterion is applicable in the given example.
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In Dummit and Foote Chapter 13: Field Theory, the authors give several examples of field extensions on page 515 - see attached.

In example (3) we read (see attached)

" (3) Take F = \mathbb{Q} and p(x) = x^2 - 2, irreducible over \mathbb{Q} by Eisenstein's Criterion, for example"

Now Eisenstein's Criterion (see other attachment - Proposition 13 and Corollary14) require the polynomial to be in R[x] where R s an integral domain.

In example (3) on page 515 of D&F we are dealing with a field, specifically \mathbb{Q}.

My problem is, then, how does Eisenstein's Criterion apply?

Can anyone please clarify this situation for me?

Peter

[This has also been posted on MHF]
 
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The sub-ring $\Bbb Z$ of $\Bbb Q$ is an integral domain...

Also, any field is automatically an integral domain. You might wish to commit to memory the following chain of inclusions:

Fields < Euclidean Domains < PID's < UFD's < Integral domains < Commutative rings.
 
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