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- Jun 22, 2012

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In Chapter 3 (Prime Ideals and Maximal Ideals) on page 44 we find Exercise 3.24 which reads as follows:

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Show that the residue class ring \(\displaystyle S \) of the ring of polynomials \(\displaystyle \mathbb{R}[x_1, x_2, x_3] \) over the real field \(\displaystyle \mathbb{R} \) in indeterminates \(\displaystyle x_1, x_2, x_3 \) given by

\(\displaystyle S = \mathbb{R}[x_1, x_2, x_3]/(x_1^2 + x_2^2 + x_3^2) \) is an integral domain.

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Can someone please help me make a significant start n this problem.

Some thoughts that I think are relevant to the problem follow:

First I tried to get an idea of the nature of \(\displaystyle (x_1^2 + x_2^2 + x_3^2) \) and \(\displaystyle S = \mathbb{R}[x_1, x_2, x_3]/(x_1^2 + x_2^2 + x_3^2) \) and their elements.

Thus ...

\(\displaystyle (x_1^2 + x_2^2 + x_3^2) = \{ f(x_1, x_2, x_3)(x_1^2 + x_2^2 + x_3^2) \ | \ f(x_1, x_2, x_3) \in \mathbb{R}[x_1, x_2, x_3] \} \)

\(\displaystyle \mathbb{R}[x_1, x_2, x_3]/(x_1^2 + x_2^2 + x_3^2) = \{ g(x_1, x_2, x_3) + (x_1^2 + x_2^2 + x_3^2) \ | \ g(x_1, x_2, x_3) \in \mathbb{R}[x_1, x_2, x_3] \} \)

Now I suspect that one now uses the division algorithm to determine a remainder which will be of lower degree than \(\displaystyle x_1^2 + x_2^2 + x_3^2 \) - that is lower than deg 2 (??? is that right ?? )

BUT, what exactly would be the nature of the remainder - ie how does the division algorithm work for polynomials of several variables?

And then ... where to go from there ...

Can someone assist me in this ...

Further does anyone know of a text that gives an example of the division algorithm, applied to polynomials in several variables ...

An elementary or undergraduate text dealing in all aspects of the theory of polynomials in several variables would be helpful in getting a sense of what is happening in the abstract theorems ...

Peter