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 and , irreducible by Eisenstein. (by Eisenstein???)
Denoting a root of p(x) by we obtain the field
with , an extension of degree 3.
To find the inverse of, say, in this field, we can proceed as follows:
By the Euclidean Algorithm in there are polynomials a(x) and b(x) with
... ... etc etc
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My problem is this:
How do D&F get the equation ?
It looks a bit like they are implying that there is a GCD of 1 between (1 + x) and 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