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Thread: 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 $\displaystyle \mathbb{R}$ shows that you are in the reals, so $\displaystyle x^2$ can have any coefficient in the reals. BUT, $\displaystyle x^2=-2$ 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 $\displaystyle x^2$ replace it with a -2. The $\displaystyle /$ 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 $\displaystyle \mathbb R[x]$ [the ring of polynomials with real coefficients] by the ideal generated by $\displaystyle x^2+2$ (is isomorphic to $\displaystyle \mathbb C)$

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


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

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

    where given a fixed $\displaystyle r\in R,$ $\displaystyle r+I$ denotes the additive coset $\displaystyle \{r+j:j\in I\}.$
    Last edited by proscientia; Dec 12th 2009 at 05:35 AM. Reason: Took a while to work out the correct homomorphism
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