Need help and confirmation about some notational symbols

What does $\displaystyle Z[X]/(x^n-1)$ mean?

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

This makes $\displaystyle (x^n-1)$ an ideal ???

And $\displaystyle Z[X]/(x^n-1)$ then is quotient ring?

Is it possible this notation to denote something other ?

Re: 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.

$\displaystyle (x^n-1)$ is not a set, but an element of Z[X].

So I'd interpret it as a set of polynomials divided by $\displaystyle (x^n-1)$:

$\displaystyle Z[X]/(x^n-1)=\{{p(x) \over x^n-1} : p(x) \in Z[X]\}$

If it is supposed to mean anything else, it is bad notation IMO and deserves a big red cross through it.

Re: 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 $\displaystyle I = (X^2 + 1)$ consisting of all multiples of the polynomial $\displaystyle X^2 + 1$. The quotient ring $\displaystyle R[X]/(X^2 + 1)$ 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" $\displaystyle X^2 + 1 = 0$, i.e. $\displaystyle X^2 = −1$, which is the defining property of i.

I'm reading and I'm reading and I'm reading and still cannot imagine how it all fits together. For example it seems that $\displaystyle X^4 = 1$ in the ring $\displaystyle Z[X]/(X^4-1)$ which seems to make sense in the context of the above citation. We "make" a ring which satisfies this property. But I cannot still imagine the cosets and their representatives...

Re: Need help and confirmation about some notational symbols

I see.

Well, a coset would be a polynomial in Z[X] plus a multiple of $\displaystyle X^4-1$.

A representative is any polynomial, with perhaps a multiple of $\displaystyle X^4-1$ subtracted, so there are no powers of 4 anymore (for easier comparison).

$\displaystyle Z[X]/(X^4-1)=\{ \{ P(X) + k(X^4-1) : k \in Z \} : P(X) \in Z[X] \}$.

Re: Need help and confirmation about some notational symbols

Thanks very much! It's getting clearer now.

Re: 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 $\displaystyle Z[X]/(X^4-1)$, which is a coset, is $\displaystyle \{ P(X) + k(X^4-1) : k \in Z \}$, where P(X) is a polynomial in Z[X].

So: $\displaystyle Z[X]/(X^4-1)=\{ \{ P(X) + k(X^4-1) : k \in Z \} : P(X) \in Z[X] \}$

Re: 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)).