Let K a field, prove:

i) is a maximal ideal in

ii) is a prime ideal in that not is maximal

Thanks!!!

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- Sep 28th 2008, 10:56 AMroporteMaximal Ideal, Prime Ideal
Let K a field, prove:

i) is a maximal ideal in

ii) is a prime ideal in that not is maximal

Thanks!!! - Sep 28th 2008, 02:39 PMThePerfectHacker
I see two ways of doing that. The first way is to let . Now consists of all expressions involving . Thus for it is necessary that for some . But any ideal which contains a unit must be the improper ideal, so . Thus, .

Another way is to form and realize that this is the consists of only constants i.e. it is isomorphic to which is a field. Therefore, must be a maximal ideal.

Quote:

ii) is a prime ideal in that not is maximal