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Let $a = \dfrac{1}{2} + i\dfrac{\sqrt{3}}{2}\in\mathbb{C}.$ My question is
show that $$\mathcal{S} = \{m + na + pa^2\,\,|\,\,m, n, p\in\mathbb{Z}\}$$ is a minimal subring which contains $a$ of ring $\mathbb{C}$

I know prove $\mathcal{S}$ is a subring of ring $\mathbb{C}$ but I don't know prove $\mathcal{S}$ is a minimal subring of ring $\mathbb{C}.$

I hope that someone can help. Thanks!

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    Isn't obvious that any subring of $\mathbb C$ containing $a$ also contains $\mathcal S$? – 2017-01-01

1 Answers 1

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If you want a hint then:

1)You have to prove that for every subring $A$ of $\mathbb{C}$ which contains $a$,we have that $S \subseteq A$.

2)Also prove that if $A$ is a subring of $\mathbb{C}$ that contains $a$ as an element and $A \subseteq S$ then $A=S$.

I think that for the minimality of $S$ you need more to prove 2)