February 26th, 2013, 02:20 PM  #1 
Member Joined: Oct 2012 Posts: 36 Thanks: 0  maximal ideal
Let F be the field and f(x)=x1,g(x)=x^21 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^21) in a prime idea domain F is maximal iff (x^21) is irreducible. And we say (x^21) is irreducible if it is not a unit, but x^21=(x+1)(x1) implies that either (x+1) or (x1) is a unit. but I can find a taylor expansion of 1/(x^21) which means (x^21) is a unit, contradicts irreducible is my idea right ??? 
February 27th, 2013, 05:06 AM  #2 
Member Joined: Jan 2013 Posts: 93 Thanks: 0  Re: Maximal ideal
You’ve probably got the right idea though I’m not sure what the Taylor expansion of has to do here. First, the statement tells you that is a twoelement 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 it’s not a field as but . 

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