# Nomenclature

• Dec 11th 2009, 02:49 PM
elninio
Nomenclature
(prove that) R[x]/<x^2+2> (is isomorphic to C)

And what exactly does the "/" imply algebraicly?

Thanks.
• Dec 11th 2009, 11:18 PM
sfspitfire23
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• Dec 11th 2009, 11:19 PM
sfspitfire23
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"
• Dec 12th 2009, 02:56 AM
Shanks
sfspitfire23 give us such a elegant explanation.
Good luck!
• Dec 12th 2009, 04:19 AM
proscientia
Quote:

Originally Posted by elninio
(prove that) R[x]/<x^2+2> (is isomorphic to C)

(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
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 $\{r+j:j\in I\}.$