Verifying Subring: Check Closure with ℂ & \alpha

[blahblah8724]

For $\alpha = (1+ \sqrt{-3})/2 \in ℂ$ and $R = \{ x +y\alpha \, | \, x,y \in Z \}$.

How would you verify that R is a subring of ℂ? Everytime I multiply two 'elements' of R to check closure I get the negative complex conjugate of $\alpha$, I think I'm doing something wrong...

Thanks!

 

[I like Serena]

For $\alpha = (1+ \sqrt{-3})/2 \in ℂ$ and $R = \{ x +y\alpha \, | \, x,y \in Z \}$.

How would you verify that R is a subring of ℂ? Everytime I multiply two 'elements' of R to check closure I get the negative complex conjugate of $\alpha$, I think I'm doing something wrong...

Thanks!

Let's see...

The negative complex conjugate of $\alpha$ is:
$$-\overline{\alpha} = - \frac 1 2 (1 - \sqrt{-3}) = \frac 1 2 (1 + \sqrt{-3}) - 1 = \alpha - 1$$

What exactly is the problem?