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GemmaJ1988

Member

Registered: 2008-10-08

Posts: 39

algebra-ideals

I have the following question:
Find all ideals in the principle ideal domain R which contain the given ideal I

Now i know this is probably a really easy question.
Is it just asking me for the rational numbers in

So in this case its just -2

Then the second part of the question is:
In which case is I the maximal ideal of R?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: algebra-ideals

Then the second part of the question is:
In which case is I the maximal ideal of R?

whatismath

Member

Registered: 2007-04-10

Posts: 19

Re: algebra-ideals

I think we don't need to apply the Euclidean Algorithm in

but just apply the fact that

is irreducible over

.

Just as Jane did. Given

. If

is an (principal) ideal,

. Write

.
Since

for some

.
But

is irreducible over

(which is just what we have known long time ago: we cannot factor

with rational coefficients),

so

is either

itself or 1. The former is rejected
since

. So

. The answers are

itself and

.

Yes

is a maximal ideal. There are infinitely many maximal ideals in

,
e.g.

.