Showing an extension field is closed under complex conjugation

Question

I want to show that the field containing $\mathbb{Q}$, and a pair of complex conjugates is closed under complex conjugation.

I'm not sure if this is even true--just a step that I think would work well for a proof I'm working on. Is this true? If so, how can I approach this?

Answer 1

I'm not sure I understand your question, but assuming we take what you've said literally:

Consider $\mathbb{Q}(i, \pi + \sqrt{2}i)$. This contains a pair of (non-real) complex conjugates, namely $\pm i$, but it does not contain the complex conjugate $\pi - \sqrt{2} i$ of $\pi + \sqrt{2}i$.