Originally Posted by

**tcRom** Show $\displaystyle Q[i + \sqrt{2}] = Q[i][\sqrt{2}]$

Now, when I do this, I find:

Basis of $\displaystyle Q[i + \sqrt{2}] = \{1, i+\sqrt{2}, 1+2i\sqrt{2}, 5i-\sqrt{2}\} $ which is equal to $\displaystyle \{1, i, \sqrt{2}, i\sqrt{2}\}$. Which is the same basis as $\displaystyle Q[i][\sqrt{2}]$.

If the basis of each field is shown to be equal, are the fields equal? It seems like an obvious yes to me, but I would like to someone else to confirm this thought if possible. I cannot find it in my text or any other resource online. Also, if someone feels like double-checking the basis for each, I would appreciate it.