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Math Help - Nomenclature

  1. #1
    Member elninio's Avatar
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    Nomenclature

    How exactly does this read:
    (prove that) R[x]/<x^2+2> (is isomorphic to C)

    And what exactly does the "/" imply algebraicly?

    Thanks.
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  2. #2
    Senior Member sfspitfire23's Avatar
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  3. #3
    Senior Member sfspitfire23's Avatar
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    Here, you are creating the complex numbers! This is very elegant. The \mathbb{R} shows that you are in the reals, so x^2 can have any coefficient in the reals. BUT, x^2=-2 There is nothing when squared in the reals equal to 2, so you have to go imaginary. So, in the multiplication table, whenever you see an x^2 replace it with a -2. The / means you are "modding out by"
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  4. #4
    Senior Member Shanks's Avatar
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    sfspitfire23 give us such a elegant explanation.
    So, follow his thread, and you will find the isomorphism.
    Good luck!
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  5. #5
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    Quote Originally Posted by elninio View Post
    How exactly does this read:
    (prove that) R[x]/<x^2+2> (is isomorphic to C)
    It reads:

    (prove that) the quotient ring of \mathbb R[x] [the ring of polynomials with real coefficients] by the ideal generated by x^2+2 (is isomorphic to \mathbb C)

    To prove this, given any f(x)\in\mathbb R[x], use the division algorithm to write f(x)=q(x)(x^2+2)+ax+b where q(x)\in\mathbb R[x] and a,b\in\mathbb R are uniquely determined by f(x). Define a map \varphi:\mathbb R[x]\to\mathbb C by \varphi\left(f(x)\right)=b+i\sqrt2a. All that remains is to show that \varphi is a ring epimomophism (surjective homomorphism) with kernel \left\langle x^2+2\right\rangle.


    Quote Originally Posted by elninio View Post
    And what exactly does the "/" imply algebraicly?
    In general if R is a ring and I is an ideal of R, R/I denotes the quotient ring of R by I:

    R/I\ =\ \{r+I:r\in R\}

    where given a fixed r\in R, r+I denotes the additive coset \{r+j:j\in I\}.
    Last edited by proscientia; December 12th 2009 at 06:35 AM. Reason: Took a while to work out the correct homomorphism
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