[FOM] Real and imaginary parts of algebraic numbers

[joeshipman at aol.com]

Clarifying question (3) -- I would like to also assume that trisections and cube roots and square roots are possible, so that I can use any operations of degree <= 4 to get from one of the two quintic inverse functions to the other. This is relevant to solving quintics using marked ruler and compass, because we already can solve cubics and quartics with these tools (in fact, with the marked ruler alone, but having a compass as well allows some higher-degree equations to be solved, and I need to know how to get from the quintics I can solve to the ones I don't yet know how to).

-- JS

-----Original Message-----
From: joeshipman <joeshipman at aol.com>
To: FOM <FOM at cs.nyu.edu>
Sent: Wed, Jul 31, 2013 6:15 pm
Subject: [FOM] Real and imaginary parts of algebraic numbers



I am working on a theory of generalized geometric constructions, which involves generating new numbers as real roots of polynomials whose coefficients are existing numbers satisfying certain relationships. The following general questions arise, and I am wondering if anyone already knows the answers or can link to a source that would have them:


(1) If x=a+bi is a root of an irreducible polynomial over the rationals of degree n, what is the maximum possible degree of the irreducible polynomials over the rationals for the real numbers a and b?


(2) If Alg_n is the field generated by all roots of polynomials over the rationals of degree ≤n, and RAlg_n is the field generated by all REAL roots of polynomials over the rationals of degree ≤n, for which n does Alg_n=RAlg_n[i]?



(3)  The inverse function of f(x)=x^5+x is uniquely defined for all real numbers; the inverse function of f(x)=x^5-x is uniquely defined for |x|>sqrt(sqrt(0.08192)) and has three real roots for |x|<sqrt(sqrt(0.08192)). Can either of these functions be obtained from the other (assuming one has access to all three real roots when they exist), using only rational operations and complex 5th roots (that is, real 5th roots and angle 5-sections)?


-- JS

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