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 $\displaystyle F = \mathbb{Q} $ and $\displaystyle p(x) = x^2 - 2 $, irreducible over $\displaystyle \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 $\displaystyle \mathbb{Q} $.

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

Can anyone please clarify this situation for me?

Peter