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Field Extensions, Polynomial Rings and Eisenstein's Criterion

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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 [TEX] F = \mathbb{Q} [/TEX] and [TEX] p(x) = x^2 - 2 [/TEX], irreducible over [TEX] \mathbb{Q} [/TEX] 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 [TEX] \mathbb{Q} [/TEX].

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

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
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.