Tower theorem

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.

Cubic curves

Dear Professor Kim,

Thank you for your reply. I have another question, this time regarding the group law on a cubic, . I am happy with showing that the points of form a group. However I am a little unsure of the ‘simplified’ group law as described in the textbook. I believe that if we take as the identity element we can treat the cubic as an affine curve. However as this corresponds to the ‘line at infinity’ in projective 2-space, what point does it correspond to in affine 2-space?
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Reply:

IF is the curve in , then it meets the line in the point as explained in the book. So we then express as a union

where We then note that
by the map

Then goes to the curve in with equation

Of course the origin does not lie on this curve, because this affine curve corresponds exactly to the complement of the origin in .

Does this answer your question? Let me know again if you’re still curious.

Question on canonical forms for quadrics

Here is a question from Dominque Miranda.

And here is my reply.