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Math Help - Need help and confirmation about some notational symbols

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    Need help and confirmation about some notational symbols

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

    Z[X] must be the ring of all polynomials with coefficients in Z.
    This makes (x^n-1) an ideal ???
    And Z[X]/(x^n-1) then is quotient ring?

    Is it possible this notation to denote something other ?
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    Super Member ILikeSerena's Avatar
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    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.

    (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 (x^n-1):
    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.
    Last edited by ILikeSerena; April 27th 2012 at 02:42 AM.
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    Re: Need help and confirmation about some notational symbols

    This is from Wikipedia article about quotient rings.

    Quotient ring - Wikipedia, the free encyclopedia

    Now consider the ring R[X] of polynomials in the variable X with real coefficients, and the ideal I = (X^2 + 1) consisting of all multiples of the polynomial X^2 + 1. The quotient ring 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" X^2 + 1 = 0, i.e. 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 X^4 = 1 in the ring 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...
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    Super Member ILikeSerena's Avatar
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    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 X^4-1.
    A representative is any polynomial, with perhaps a multiple of X^4-1 subtracted, so there are no powers of 4 anymore (for easier comparison).

    Z[X]/(X^4-1)=\{ \{ P(X) + k(X^4-1) : k \in Z \} : P(X) \in Z[X] \}.
    Last edited by ILikeSerena; April 28th 2012 at 02:14 AM.
    Thanks from mrproper
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    Re: Need help and confirmation about some notational symbols

    Thanks very much! It's getting clearer now.
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    Super Member ILikeSerena's Avatar
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    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 Z[X]/(X^4-1), which is a coset, is \{ P(X) + k(X^4-1) : k \in Z \}, where P(X) is a polynomial in Z[X].

    So: Z[X]/(X^4-1)=\{ \{ P(X) + k(X^4-1) : k \in Z \} : P(X) \in Z[X] \}
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    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) = x2 + 1. note in this case, if we write I = (x2 + 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) = (x2 + I) + (1 + I) = (x2 + 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)).
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