Hi Chiroptera,
The polynomial is of degree 3, which is not a power of 2.
Uh-huh, with you so far.
If an irrational number is not a root of a irreducible polynomial of degree a power of 2...
Very interesting, but what's that got to do with the case at hand? All you know is it's a root of an irreducible polynomial of degree 3. A bit missing here I think.
I got the impression that Percy's incredulity at the impossibility of trisecting an angle (albeit he misunderstaood the proof) stemmed from it somehow being proven that there couldn't be "some other way of doing it" (whether taking a cube root or trisecting an angle). How could you prove there's no other way?
Obviously this has been proven, as you show. However, step 10 is one hell of a leap! So I'm not sure it will help Percy much unless we could show how constructible lengths relate to polynomials (of any kind). I'm not saying I can do this - it's been > 20 years.
Lastly, I think you may have over complicated things. The point is that a number which is the root of a polynomial of degree 2 or less is constructible. No need to mention irrationals or irreducible polynomials (well, not at step 10 at least).
If I knew how to do them, there'd be smiley faces above - not trying (but possibly succeeding) to be a pain in the arse with criticisms, simply hoped to clarify.
It's been too long since I've had algebra.
Such a shame, you're missing out on all the fun. What do they make you do these days?
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