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- Jun 22, 2012

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I have a question regarding the nature of extension fields.

Theorem 4 (D&F Section 13.1, page 513) states the following (see attachment):

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Theorem 4. Let [TEX] p(x) \in F[x] [/TEX] be an irreducible polynomial of degree n over a field F and let K be the field [TEX] F[x]/(p(x)) [/TEX]. Let [TEX] \theta = x \ mod \ (p(x)) \in K [/TEX]. Then the elements

[TEX] 1, \theta, {\theta}^2, ... ... , {\theta}^{n-1} [/TEX]

are a basis for K as a vector space over F, so the degree of the extension is n i.e.

[TEX] [K \ : \ F] = n [/TEX]. Hence

[TEX] K = \{ a_0 + a_1 \theta + a_2 {\theta}^2 + ... ... + a_{n-1} {\theta}^{n-1} \ | \ a_0, a_1, ... ... , a_{n-1} \in F \} [/TEX]

consists of all polynomials of degree [TEX] \lt n [/TEX] in [TEX] \theta [/TEX]

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However, when we come to Example 4 on page 515 of D&F we read the following: (see attachment)

(4) Let [TEX] F = \mathbb{Q} [/TEX] and [TEX] p(x) = x^3 - 2 [/TEX] which is irreducible by Eisenstein.

Denoting a root of p(x) by [TEX] \theta [/TEX] we obtain the field

[TEX] \mathbb{Q}[x]/(x^3 - 2) \cong \{a + b \theta + c {\theta}^2 \ | \ a, b, c \in \mathbb{Q} [/TEX]

with [TEX] {\theta}^3 = 2 [/TEX] an extension of degree 3. ... ... etc

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Now my problem is that in Theorem 4 we read

[TEX] K = \{ a_0 + a_1 \theta + a_2 {\theta}^2 + ... ... + a_{n-1} {\theta}^{n-1} \ | \ a_0, a_1, ... ... , a_{n-1} \in F \} [/TEX] which becomes

[TEX] K = \{a + b \theta + c {\theta}^2 [/TEX] in the situation of Example 4

But then in Example 4 we have

[TEX] K = \mathbb{Q}[x]/(x^3 - 2) \cong \{a + b \theta + c {\theta}^2 \ | \ a, b, c \in \mathbb{Q} [/TEX]

???

It seems that in Theorem 4, we have [TEX] \theta = x \ mod \ (p(x)) [/TEX] but in Example (4) we have [TEX] \theta = \sqrt[3]{2} [/TEX] and we do not have equality but only an isomorphism, that is [TEX] \mathbb{Q}[x]/(x^3 - 2) \cong \mathbb{Q}(\sqrt[3]{2} [/TEX].

In Field theory we seem to prove that an irreducible polynomial has a root in a field that is isomorphic to the actual field that contains the root.

Does what I am saying make sense? Can someone clarify this issue for me?

Peter

**Notes:**

1. I think Deveno was trying to clarify this for me in a previous post but since I was not quite sure of things I am trying to further clarify the issue

2. The above has also been posted on MHF