What does mean?

Z[X] must be the ring of all polynomials with coefficients in Z.

This makes an ideal ???

And then is quotient ring?

Is it possible this notation to denote something other ?

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- Apr 26th 2012, 09:26 AMmrproperNeed help and confirmation about some notational symbols
What does mean?

Z[X] must be the ring of all polynomials with coefficients in Z.

This makes an ideal ???

And then is quotient ring?

Is it possible this notation to denote something other ? - Apr 27th 2012, 01:39 AMILikeSerenaRe: Need help and confirmation about some notational symbols
An ideal is a set.

The denominator of a quotient ring also has to be a set.

is not a set, but an element of Z[X].

So I'd interpret it as a set of polynomials divided by :

If it is supposed to mean anything else, it is bad notation IMO and deserves a big red cross through it. - Apr 27th 2012, 02:36 AMmrproperRe: Need help and confirmation about some notational symbols
This is from Wikipedia article about quotient rings.

Quotient ring - Wikipedia, the free encyclopedia

Quote:

Now consider the ring R[X] of polynomials in the variable X with real coefficients, and the ideal consisting of all multiples of the polynomial . The quotient ring is naturally isomorphic to the field of complex numbers C, with the class [X] playing the role of the imaginary unit i. The reason: we "forced" , i.e. , which is the defining property of i.

- Apr 27th 2012, 01:31 PMILikeSerenaRe: Need help and confirmation about some notational symbols
I see.

Well, a coset would be a polynomial in Z[X] plus a multiple of .

A representative is any polynomial, with perhaps a multiple of subtracted, so there are no powers of 4 anymore (for easier comparison).

. - Apr 27th 2012, 10:44 PMmrproperRe: Need help and confirmation about some notational symbols
Thanks very much! It's getting clearer now.

- Apr 28th 2012, 01:13 AMILikeSerenaRe: Need help and confirmation about some notational symbols
I just saw I need to modify my equation.

I've edited it in my previous post.

An element of , which is a coset, is , where P(X) is a polynomial in Z[X].

So: - Apr 28th 2012, 02:46 PMDevenoRe: Need help and confirmation about some notational symbols
the notation (f(x)) is often used for the principal ideal generated by f(x), that is: all polynomials of the form k(x)f(x), for k(x) in Z[x].

note that the cosets p(x) + (f(x)) consist of the cosets of those polynomials p(x) where deg(p(x)) < deg(f(x)) (since we can use the division algorithm to write

p(x) = q(x)f(x) + r(x) in which case p(x) + (f(x)) = [r(x) + (f(x))] + [q(x)f(x) + (f(x))] = r(x) + (f(x)) since q(x)f(x) is in (f(x)) so that q(x)f(x) + (f(x)) = 0 + (f(x)) = (f(x)) itself).

the "classic example" is where f(x) = x^{2}+ 1. note in this case, if we write I = (x^{2}+ 1), in Z[x]/I we have cosets of the form:

a + bx + I, where a,b are integers. also note that for the coset x + I:

(x + I)(x + I) + (1 + I) = (x^{2}+ I) + (1 + I) = (x^{2}+ 1) + I = I, the 0-element of Z[x]/I.

thus a + bx + I ↔ a + bi is a (ring) isomorphism of Z[x]/I with Z[i], the gaussian integers (replacing "polynomials in x" with "polynomials in i" (i being a square root of -1)).