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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- December 11th 2009, 01:49 PMelninioNomenclature
How exactly does this read:

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

And what exactly does the "/" imply algebraicly?

Thanks. - December 11th 2009, 10:18 PMsfspitfire23
...

- December 11th 2009, 10:19 PMsfspitfire23
Here, you are creating the complex numbers! This is very elegant. The shows that you are in the reals, so can have any coefficient in the reals. BUT, 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 replace it with a -2. The means you are "modding out by"

- December 12th 2009, 01:56 AMShanks
sfspitfire23 give us such a elegant explanation.

So, follow his thread, and you will find the isomorphism.

Good luck! - December 12th 2009, 03:19 AMproscientia
It reads:

(prove that) the quotient ring of [the ring of polynomials with real coefficients] by the ideal generated by (is isomorphic to

To prove this, given any use the division algorithm to write where and are uniquely determined by Define a map by All that remains is to show that is a ring epimomophism (surjective homomorphism) with kernel

In general if is a ring and is an ideal of denotes the quotient ring of by :

where given a fixed denotes the additive coset