I am reading Nicholson: Introduction to Abstract Algebra, Section 6.3 Splitting Fields.
Example 1 reads as follows: (see attachment)
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Example 1. Find an extension in which factors completely into linear factors.
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The solution reads as follows:
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Solution. The polynomial f(x) is irreducible over (it has no root in ) so
is a field containing a root t of f(x).
Hence x + t = x - t is a factor of f(x)
The division algorithm gives where
, so it suffices to show that g(x) also factors completely in E.
Trial and error give so for some .
... ... etc (see attachment)
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My problem is that I cannot show how implies that for some .
I would appreciate some help.
Peter