I am reading Nicholson: Introduction to Abstract Algebra, Section 6.3 Splitting Fields.

Example 1 reads as follows: (see attachment)

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Find an extension in which factors completely into linear factors.Example 1.

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The solution reads as follows:

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The polynomial f(x) is irreducible over (it has no root in ) soSolution.

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