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- Jun 22, 2012

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I am reading Dummit and Foote Ch 13 on Field Theory.

On page 515-516 D&F give a series of basic examples on field extensions - see attachment.

The start to Example (4) reads as follows: (see attachment)

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

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

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

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

To find the inverse of, say, [TEX] 1 + \theta [/TEX] in this field, we can proceed as follows:

By the Euclidean Algorithm in [TEX] \mathbb{Q}[x] [/TEX] there are polynomials a(x) and b(x) with

[TEX] a(x)(1 + x) + b(x)(x^3 - 2) = 1 [/TEX]

... ... etc etc

-----------------------------------------------------------------------------------------

My problem is this:

How do D&F get the equation [TEX] a(x)(1 + x) + b(x)(x^3 - 2) = 1 [/TEX]?

It looks a bit like they are implying that there is a GCD of 1 between (1 + x) and [TEX] (x^3 - 2) [/TEX] and then use Theorem 4 on page 275 (see attached) relating the Euclidean Algorithm and the GCD of two elements of a Euclidean Domain, but I am not sure and further, not sure why the GCD is 1 anyway.

Can someone please clarify the above for me?

Peter

[This has also been posted on MHF]

On page 515-516 D&F give a series of basic examples on field extensions - see attachment.

The start to Example (4) reads as follows: (see attachment)

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

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

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

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

To find the inverse of, say, [TEX] 1 + \theta [/TEX] in this field, we can proceed as follows:

By the Euclidean Algorithm in [TEX] \mathbb{Q}[x] [/TEX] there are polynomials a(x) and b(x) with

[TEX] a(x)(1 + x) + b(x)(x^3 - 2) = 1 [/TEX]

... ... etc etc

-----------------------------------------------------------------------------------------

My problem is this:

How do D&F get the equation [TEX] a(x)(1 + x) + b(x)(x^3 - 2) = 1 [/TEX]?

It looks a bit like they are implying that there is a GCD of 1 between (1 + x) and [TEX] (x^3 - 2) [/TEX] and then use Theorem 4 on page 275 (see attached) relating the Euclidean Algorithm and the GCD of two elements of a Euclidean Domain, but I am not sure and further, not sure why the GCD is 1 anyway.

Can someone please clarify the above for me?

Peter

[This has also been posted on MHF]

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