Originally Posted by
elninio 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 $\displaystyle \left\langle x^2+2\right\rangle.$
Originally Posted by
elninio 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 $\displaystyle \{r+j:j\in I\}.$