MHB Field Extensions, Polynomial Rings and Eisenstein's Criterion

[Math Amateur]

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]

 

[Deveno]

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.