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February 26th, 2013, 02:20 PM   #1
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maximal ideal

Let F be the field and f(x)=x-1,g(x)=x^2-1 and F[x]/(f(x)) is isomorphism to F, is it g(x) maximal??

I will say no.Since g(x) is not 0, the dieal (x^2-1) in a prime idea domain F is maximal iff (x^2-1) is irreducible.
And we say (x^2-1) is irreducible if it is not a unit, but x^2-1=(x+1)(x-1) implies that either (x+1) or (x-1) is a unit.
but I can find a taylor expansion of 1/(x^2-1) which means (x^2-1) is a unit, contradicts irreducible

is my idea right ???
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February 27th, 2013, 05:06 AM   #2
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Re: Maximal ideal

Youíve probably got the right idea though Iím not sure what the Taylor expansion of has to do here.

First, the statement tells you that is a two-element field since the elements of are and . So the elements of are , , , . (NB: in the field .)

Now is maximal in if and only if is a field. But itís not a field as but .
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