# Thread: Field Extensions - Basic Theory

1. ## Field Extensions - Basic Theory

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 $\displaystyle F = \mathbb{Q}$ and $\displaystyle p(x) = x^3 - 2$, irreducible by Eisenstein. (by Eisenstein???)

Denoting a root of p(x) by $\displaystyle \theta$ we obtain the field

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

with $\displaystyle {\theta}^3 = 2$ , an extension of degree 3.

To find the inverse of, say, $\displaystyle 1 + \theta$ in this field, we can proceed as follows:

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

$\displaystyle a(x)(1 + x) + b(x)(x^3 - 2) = 1$

... ... etc etc

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My problem is this:

How do D&F get the equation $\displaystyle a(x)(1 + x) + b(x)(x^3 - 2) = 1$?

It looks a bit like they are implying that there is a GCD of 1 between (1 + x) and $\displaystyle (x^3 - 2)$ 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