Field Extensions, Polynomial Rings and Eisenstein's Criterion

In summary, the conversation discusses field extensions and the application of Eisenstein's Criterion in an example involving the field $\mathbb{Q}$. The question is raised about how Eisenstein's Criterion applies in this situation, to which it is clarified that any field is automatically an integral domain.
  • #1
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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 [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]
 
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  • #2
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.
 

FAQ: Field Extensions, Polynomial Rings and Eisenstein's Criterion

1. What is a field extension?

A field extension is a mathematical concept that involves extending a field (a mathematical structure with addition, subtraction, multiplication, and division operations) by adding new elements to it. These new elements are called "extensions" and are usually created by adding roots of polynomials to the original field.

2. How are field extensions related to polynomial rings?

Field extensions and polynomial rings are closely related because field extensions are often created by adding roots of polynomials to the original field. These roots are then used to create new elements in the polynomial ring, which is a mathematical structure that allows for the manipulation and study of polynomials.

3. What is Eisenstein's Criterion?

Eisenstein's Criterion is a method for determining whether a given polynomial is irreducible (cannot be factored) over a given field. It states that if a polynomial has a certain form (specifically, if the coefficients are all integers and the constant term is not divisible by a prime number, while all other coefficients are divisible by that prime), then the polynomial is irreducible over that field.

4. How is Eisenstein's Criterion used in field extensions?

Eisenstein's Criterion is often used in field extensions to determine whether a given polynomial can be factored into smaller polynomials over the original field. If the polynomial is irreducible over the original field, then it cannot be factored into smaller polynomials and thus cannot be used to create new elements for the field extension.

5. Can Eisenstein's Criterion be used for all polynomials?

No, Eisenstein's Criterion can only be used for certain types of polynomials that meet the specific criteria outlined in the criterion. If a polynomial does not meet these criteria, then a different method must be used to determine its irreducibility over a given field.

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