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- #1

- Jun 22, 2012

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Example 1 reads as follows: (see attachment)

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*Find an extension [TEX] E \supseteq \mathbb{Z}_2 [/TEX] in which [TEX] f(x) = x^3 + x + 1 [/TEX] factors completely into linear factors.*

**Example 1.**--------------------------------------------------------------------------------------------------

The solution reads as follows:

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*The polynomial f(x) is irreducible over [TEX] \mathbb{Z}_2 [/TEX] (it has no root in [TEX] \mathbb{Z}_2 [/TEX] ) so*

**Solution.**[TEX] E = \{ a_0 + a_1 t + a_2 t^2 \ | \ a_i \in \mathbb{Z}_2 , f(t) = 0 \} [/TEX]

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 [TEX] f(x) = (x+t) g(x) [/TEX] where [TEX] g(x) = x^2 + tx + (1 + t^2) [/TEX]

, so it suffices to show that g(x) also factors completely in E.

Trial and error give [TEX] g(t^2) = 0 [/TEX] so [TEX] g(x) = (x + t^2)(x + v) [/TEX] for some [TEX] v \in F[/TEX].

... ... etc (see attachment)

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**My problem is that I cannot show how [TEX] g(t^2) = 0 [/TEX] implies that [TEX] g(x) = (x + t^2)(x + v) [/TEX] for some [TEX] v \in F[/TEX].**I would appreciate some help.

Peter

[Note; This has also been posted on MHF]