Dear Prof Kim

I can show that . But not 8. Why is that?

Can you explain why and give me instruction to prove it please?

Thanks very much.

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Reply: It’s important at this point to understand precisely what goes into the tower theorem.

For the first case, we have

But the important point is that the square root we’re throwing in at each stage is not in the previous field. That is, and . Of course, these statements must be proved.(Try it!)

Now let’s examine the tower

In this case, it’s true that , so that the degree of

is 4. But note that

and . So in fact,

This explains the degree 4.