Subject | Some corrections |
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Date | Tue, 30 Nov 2000 11:58:44 -0800 (PST) |

By | Ed Fisher |

Alex,

Since I like your site so much, I am inspired to offer some corrections/suggestions.

The concept of "trisectable" is not like "constructible" in the sense that you suggest. Indeed, if two angles are constructible, then their sum or difference is also, SINCE YOU HAVE BOTH ANGLES IN HAND TO WORK WITH. However, if two angles are trisectable, then I see no reason why their sum or difference need be, since, given such a sum (say), you cannot assume that you have the original two angles IN HAND (since they are not necessarily constructible, even from this sum). Hence, you cannot use the two original angles and, by trisecting them, arrive at a trisection of the sum. Thus, I find the following Description misleading:

"The argument is completely general. Let A be a property of angles, such as being constructible or being trisectable. Then, schematically,

A - A = A

which simply says that the difference of two angles with property A also possesses that property. (Which is of course true for A being either constructible or trisectable.)"

In the specific example, π/3 is actually constructible (from nothing), so you cannot construct from it π/21 or 2π/7 in order to do a trisection for π/3 from the ones you (hypothetically) have for them. In other words, I do not see that this argument can be made to work in the example either.

On the other hand, is u a primitive 7^{th} root of unity, then u^{3} is also
a primitive 7^{th} root of unity (since ^{3})^{5} = (u^{14})×u = u.^{3}
is not constructible, while the field it generates over the constructibles contains its cube root. This was the first part of your piece. [In fact, you can see that u itself has a relatively constructible cube root, viz., u^{5}, since ^{3})^{5} = (u^{14})×u = u.^{nd} root of unity (angle π/21), the field generated by it has degree 12 over the rationals, while its cube root (being a primitive 126^{th} root of unity) has degree 36 over the rationals. [The theorem is that any primitive nth root of unity has degree

I hope I have been clear on this matter. Sorry it rambles on so much. Feel free to ask about anything that was not clear.

Appreciatively,

Ed Fisher

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