nhaliday + math.cv   17

Rational Sines of Rational Multiples of p
For which rational multiples of p is the sine rational? We have the three trivial cases
[0, pi/2, pi/6]
and we wish to show that these are essentially the only distinct rational sines of rational multiples of p.

The assertion about rational sines of rational multiples of p follows from two fundamental lemmas. The first is

Lemma 1: For any rational number q the value of sin(qp) is a root of a monic polynomial with integer coefficients.

[Pf uses some ideas unfamiliar to me: similarity parameter of Moebius (linear fraction) transformations, and finding a polynomial for a desired root by constructing a Moebius transformation with a finite period.]

...

Lemma 2: Any root of a monic polynomial f(x) with integer coefficients must either be an integer or irrational.

[Gauss's Lemma, cf Dummit-Foote.]

...
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july 2019 by nhaliday
cv.complex variables - Absolute value inequality for complex numbers - MathOverflow
In general, once you've proven an inequality like this in R it holds automatically in any Euclidean space (including C) by averaging over projections. ("Inequality like this" = inequality where every term is the length of some linear combination of variable vectors in the space; here the vectors are a, b, c).

I learned this trick at MOP 30+ years ago, and don't know or remember who discovered it.
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january 2017 by nhaliday
ca.analysis and odes - Why do functions in complex analysis behave so well? (as opposed to functions in real analysis) - MathOverflow
Well, real-valued analytic functions are just as rigid as their complex-valued counterparts. The true question is why complex smooth (or complex differentiable) functions are automatically complex analytic, whilst real smooth (or real differentiable) functions need not be real analytic.
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january 2017 by nhaliday