Re: Prime ideals of C[x]?
From: KRamsay (kramsay_at_aol.com)
Date: 10/12/04
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Date: 12 Oct 2004 05:12:06 GMT
In article <ckfhdk$cur2$1@netnews.upenn.edu>, "Isaac" <Isharu@yahoo.com>
writes:
|I can see that C[x] / (x^m) is isomorphic to just polynomials of degree at
|most m-1 with coefficients in C.
Well, that's a little imprecise. The polynomials of degree at most m-1
aren't closed under multiplication. Each element of C[x]/(x^m) is a
set of the form P+(x^m). Each such set has a representative that's
a polynomial of degree <=m-1 (subtract off the terms in P of degree
>=m).
| But what are ideals of this ring? What
|are prime ideals in this ring?
Start with a simple special case. What are the ideals of C[x]/(x^2)
or C[x]/(x^3)? The set {0} is an ideal of C[x]/(x^2), more often
written as (0). One hint I can give you is that a lot of the elements
of each of these are units. The ring generated by a single polynomial
can be simplified by multiplying the polynomial by another one that
makes the result (reduced mod x^2 or x^3 and so on) simple.
Remember the definition of "prime ideal". If a product of two elements
ab is in the ideal, then one of a or b is in the ideal. By induction, if
some product a1...an is in the ideal, one of the a_i is. Notice that
each ideal in C[x]/(x^n) contains 0 and in C[x]/(x^n) we have x^n=0.
|Related to this is C[x] has a prime ideal I know, namely (x), since C is an
|integral domain. Is that the only prime ideal of C[x]?
No, (x-1) is another prime ideal.
|So I'm also trying to understand ideals in C[x]. For instance, is there
|anything different between (x^2) and (x^2+x^4)? I'm not sure. Firstly,
|x^2+x^4 = x^2(x^2+1), so every element in the second ideal can be written as
|an element in the first ideal, so clearly (x^2+x^4) is contained in (x^2).
|However, (x^2) I am not sure if it is contained in (x^2+x^4). In
|particular, x^2 isn't in (x^2+x^4), right?
Yes. Remember the definition of (a) where a is an element of a ring.
It's the set of elements of the form ax where x is an element of the
ring. (I'm assuming your rings are rings with 1.) So the question is
whether there exists a polynomial P(x) with coefficients in C, such
that x^2=(x^2+x^4)*P.
|But if this is the case, what is
|the difference between C[x] / (x^2) and C[x] / (x^2+x^4) ??
For one thing, the first ring is a two-dimensional vector space
over C, and the second ring is a four-dimensional vector
space over C.
Do you know what the rings Z/(m) are like, where m is an
integer? Did you know that they can sometimes be written
as products of simpler rings, e.g. Z/(6) is isomorphic to
Z/(2) x Z/(3) by the function 1->(1,1). This works out smoothly
because the ideals in Z are all of the form (m) and Z has
unique factorization. The situation in C[x] is parallel, and
is well worth understanding well.
Keith Ramsay
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