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**HTale** Hi guys,

I'm scheduled to see my lecturer about this problem, but he isn't available until next week, and I feel that I am likely to get a clearer explanation here.

This is for a Galois Theory course, and he states the following theorem,

Let $\displaystyle (S, +, \times)$ be a subring of $\displaystyle (R, +, \times)$ and fix $\displaystyle r \in R$. Then the smallest subring of $\displaystyle (R,+,\times)$ that contains both $\displaystyle S$ and $\displaystyle r$ is equal to

$\displaystyle \displaystyle \{ \sum^{i=n}_{i=0} s_ir^i : s_i \in S, n \in \mathbb{N}\}$

and then he writes this subring as $\displaystyle S[r]$.

Then, he writes an example, $\displaystyle S[r] = \mathbb{Q}[i]$, and concludes that

$\displaystyle \mathbb{Q}[i] = \displaystyle \{ r_0 + ir_1 : r_0, r_1 \in \mathbb{Q}\}$

This is fine. The problem is when he introduces subrings of two elements. He claims