Minimal Polynomials of Algebraic Numbers with Rational Parameters

It is unusual for an irreducible polynomial to have a root with rational real part or with rational imaginary part. Of course, such polynomials exist: one can simply take the minimal polynomial of, say, 1+ i √ 2 or √ 2+ i. The same applies to polynomials having a root of rational modulus. But it turns out to be of interest to characterize these three kinds of polynomials. We therefore define our first family of polynomials, C1, to consist of the minimal polynomials of some algebraic number having rational real part. Our second family, C2, consists of the minimal polynomials of some algebraic number having rational imaginary part, while our third family, C3, consists of the minimal polynomials of some algebraic number having rational modulus . We describe the polynomials of each family in Sections 3, 4 and 5. Then in Sections 6, 7 and 8 we classify the polynomials belonging to two of the three families, while in Section 9 we do the same for the polynomials belonging to all three families. Section 2 contains preliminary results needed for the proofs. For a rational linear polynomial, its root has rational real part, imaginary part and modulus. An irreducible quadratic polynomial z + pz + q with rational coefficients and with discriminant ∆ = p− 4q belongs to C1 if and only if ∆ < 0, to C2 if and only if −∆ is a square (in which case it belongs to C1 ∩ C2) or ∆ > 0, and to C3 if and only if ∆ < 0 and q is a square (in which case it belongs to C1 ∩ C3). Hence it belongs to C2 ∩ C3 if and only if −∆ and q are both squares (in which case it belongs to C1 ∩ C2 ∩ C3). For −∆ = a and q = b this latter condition is 4b − a = p. This is essentially the Pythagoras equation, with solution p = u−v, a = 2uv and 2b = u+v for some rationals u and v. For the rest of the paper, therefore, we can restrict our attention to polynomials of degree at least 3.